The prisoners for the Alexander were ordered to be embarked on the ship from Newgate prison in January 1787 -
(Extract) Under Secretary Nepean to Mr. Shelton, Whitehall, 1 January 1787, I understand from Mr. Campbell that you have already been furnished with a list of such convicts as are to be sent out in the Alexander, and as it is wished that they may be removed as soon as possible from the hulk to make room for the people now in Newgate, I will beg of you to get the bonds and contracts executed with a little delay as may be. The Lady Penrhyn now in the river, will receive all the female convicts now in Newgate under sentence of transportation and her commander with Mr. Richardson, will enter into the bonds the latter will contract, There will be a further list sent to you, I hope today of other convicts now in the hulks who are to be sent to the ship Scarborough at Portsmouth, to be conveyed from thence to the intended settlement at Botany Bay, and which will include all that are to go out in the first convoy. 
Governor Phillip's correspondence to Under Secretary Nepean reveals that the Alexander was overcrowded while still in the river and some of the convicts were already ill -
London, Jan. 11, 1787 Sir, By letters from Lieutenant Shortland and the surgeon's assistant on board the Alexander, I find that 184 men are put on board that ship, and 56 women on board the Lady Penrhyn, that there are amongst the men several unable to help themselves, and that no kind of surgeon's instruments have been put on board that ship or any of the transports. You will, sir, permit me to observe that it will be very difficult to prevent the most fatal sickness amongst men so closely confined; that on board that ship which is to receive 210 convicts there is not a space left for them to move in sufficiently large for 40 men to be in motion at the same time, nor is it safe to permit any number of men to be on deck while the ship remains so near the land. On this consideration, I hope that you will order the Alexander and Lady Penrhyn to join his Majesty's ship Sirius immediately and proceed to Spithead, where more liberty may be allowed the convicts than can be done with safety in the river, and those that are wanting to compleat the number to be sent out in those ships may be sent round to Portsmouth with the other convicts, for the most fatal consequences may be expected if the full number is kept on board any length of time before we sail. You will, I presume, see the necessity of ordering some surgeon's instruments to be sent on board the ships that carry the convicts - and I do hope that no more will be embarked till the ships are ready to sail, and which they cannot be for a week after they get to Spithead.
The detachment of Marines on board consisted of two First Lieutenants - Lieutenant John Johnstone and Lieutenant James Maitland Shairp; three sergeants, two corporals, one drummer, 28 privates, one woman making a total of 37 persons.
Major Ross reported of the terrible sickness amongst the Marines on the Alexander in correspondence to Secretary Stephens dated 13th April 1787 at Portsmouth
I have to request you will please to inform their Lordships that the sickness which has, and still does prevail among the marine detachment embarked on board the Alexander, gives me a great degree of concern. Since the time of their first embarkation no less than one serjeant, one drummer, and fourteen privates have been sent sick on shore from her, some of whom I am informed, are since dead.
How far the cause of this sickness may be of a malignant kind is not for me to say, but I am apt to suppose that the situation in which the marines are placed in that ship is of itself sufficient to cause the disorder with which they are afflicted. I beg leave to observe for their Lordships information that the place allotted for the marines on board the Alexander is upon a deck, and immediately under where the seamen are birthed. In this place they are excluded from all air, but what passes through the hatchway leading from the seamen's berth, which must in some degree render it putrid before it reaches the others.
In the same correspondence Major Ross also told of the dire situation of the families of the Marines after he found on visiting the ships at the Motherbank that a Marine, his wife and two children all were subsisting on only a ration and a half of provisions
The eleven vessels of the First Fleet departed from Portsmouth early on Sunday 13 May 1787.
Extract of a letter from an Officer on board one of the ships destined for Botany Bay dated Tenerife, June 5 1787........
After a very pleasant passage of exactly three weeks, we arrived here on Sunday evening the 3rd July The fleet has been throughout exceedingly healthy and still continues so, having lost only five convicts since we sailed. In most of the ships these unhappy men have been released from their irons, and their behaviour since has evinced the propriety of this indulgence; besides which during our stay here, every individual of the squadron has partaken amply of the good things which this sterile unyielding soil produces; fresh beef and vegetables (as far as it has been possible to procure them) have been universally extended, so that we have every reasonable prospect of continuing in our present healthy state for a length of time to come. Should nothing unexpected intervene, we shall take our departure for Rio de Janeiro, in the Brazils, on the 8th of this month and very possibly touch for a day or two at St. Jago, in running between Cape de Verde Islands; but this last I do not speak of with confidence on the contrary, I think the commodore will push for Rio with all possible expedition it being a part of all others which affords those refreshments so absolutely necessary in a long sea voyage. - I had almost forgot to tell you, that Commodore Phillips this morning, introduced twenty officers which were all that could be spared on shore at once, to the Marquis de Branciforte, the Spanish Governor of this island; in the number was your humble servant; His Excellency was pleased to receive us with that politeness and affability which the inhabitants of Teneriffe, say at all times distinguish his behaviour. 
Rio de Janeiro
They arrived at Rio de Janeiro on 6th August 1787 and sailed from there on 5th September 1787.
In correspondence from an Officer on one of the vessels the following account was reported in the Freeman's Journal..... On the 7th October the Alexander transport threw signals of distress, upon which the Commodore ordered our boats to be manned and sent on board. The Captain of the Alexander informed the commanding officer, that his men had mutinied, and attempted to release the convicts in order to strengthen themselves; but our well times assistance prevented these desperadoes from effecting their evil intention. Our people secured four of the most daring, and sent them in irons on board the Sirius. The principal part of the marines on board the Alexander being ill, and therefore unable to prevent any attempts they might make after our departure, the next day the Alexander sent two convicts on board our ship, they having been very disorderly the preceding night.
Cape of Good Hope
The Fleet reached the Cape of Good Hope on 13th October 1787.
The Alexander arrived in Botany Bay on 19 January 1788.
Departure From the Colony
The Alexander departed in July 1788 in company with the Prince of Wales, Borrowdale and Friendship transports under direction and command of Lieut. John Shortland (senior), Agent for Transports.
These ships were all in a distressed state when they sailed, both as to sickness and want of provisions. The Alexander was to load a cargo of tea at Canton for the East India Company.
It was unanimously agreed that the vessels should sail northward through Torres Straits as the season was too advanced for them to attempt the southern course. On the morning of July 11th, 1788, Lieut. Shortland, in the Alexander, sailed from Port Jackson and outside the Heads joined company with the other three transports. Intending to touch at Lord Howe Island, the course directed was E.N.E., but unfortunately the vessels had no instructions where to rendezvous in case of separation. When the vessels cleared the port they were compelled to carry a press of sail in order to get off the coast, as they were very light and there was a powerful southerly current setting in upon the shore. They experienced fresh gales with rain from the south east for two days, which with the heavy swell, made it very difficult to keep clear of the shore, and that shore was within the neighbourhood of the coast of Newcastle. When off Port Stephens Lieutenant Shortland, in his journal, wrote: 'About 8am on the 16th July, 1788, the rocks off the entrance of Port Stephens bore N.W. by W., distant three leagues. He then writes that he regretted very much that this place has not been surveyed; had it been known to afford safe anchorage, it would have been much more prudent to have put in there and wait for a change of wind than to attempt keeping the sea in circumstances so very unfavourable with ships so little calculated to run along so great extent of lee shore.
When off the port he fired guns and made signal for the vessels to rendezvous at Lord Howe Island, but some of them failed to see the signals. The Prince of Wales and Borrowdale were seen no more by Lieut. Shortland throughout the voyage they having taken the southern course round Cape Horn to Rio de Janeiro. 
* In 1797 Lieut. Shortland's son Lieutenant John Shortland (junior) discovered the Hunter River. He produced an eye sketch of the mouth of the river and coast line which would have been particular interest to his father who had sailed there almost a decade before.
The following details of the remainder of the voyage of the Alexander home are included in the Memoirs of Hydrography......The Middleton reef and island New Georgia and Shortland of the Solomon islands the Treasury Four and Wallis islands were discovered Pellew islands were communicated with and the north east Coast of Borneo reached October 17th. Here the crews of the vessels having been terribly reduced by scurvy the Friendship was sunk and the remnant of her ships company transferred to the Alexander. Continuing to Batavia the Alexander arrived November 19th in an almost helpless state only one man besides the officers being able to work aloft. A new crew was here embarked and at the Cape of Good Hope the Sirius Captain Hunter fallen in with, who gave Lieutenant Shortland information that the missing transports Borrowdale and Prince of Wales had returned to England by the southern passage The Alexander arrived at the Isle of Wight May 28th 1789.
The next convict ship to arrive in New South Wales after the First Fleet was the Lady Juliana.
Notes and Links
1). Francis Fowkes, a former navy midshipman arrived as a convict on the Alexander having been convicted of stealing a coat and boots.
The National Library.......The first known printed map of the colony is thought to have been drawn by Fowkes three months after the arrival of the first fleet. The map shows the beginnings of the settlement at Port Jackson, including the eleven ships of the First Fleet and the buildings, tents, sawpits, workshops, storehouses and gardens of the First Settlement.
2). John Wilson, wild white man and explorer, was convicted in October 1785, at Wigan, Lancashire, England, and arrived as a convict on the Alexander.
4). Richard Clough arrived as a convict on the Alexander. He died in 1793 and was buried in the Old Sydney Burial Ground as was George Robinson who died in November 1793, John Stogdell in March 1801 and James Welch who died in December 1792
5). Read the story of Robert Abel, one of the convicts on the Alexander at London Lives
6). NEWCASTLE'S PIONEER. A STORY OF HIS BROTHER. 'In the interesting article on 'Newcastle's Pioneer Explorer,' in the Supplement to the 'Evening News' of March 22, the writer (says a correspondent) briefly mentions that a brother, of the pioneer, Thomas George Shortland, was a midship man on one of the vessels of the 'First Fleet.' Permit me to add that this officer afterwards, like his brother, greatly distinguished himself. On August 4, 1798. five boats from the 38-gun frigate Melpomene and 14-gun brig-sloop Childers under the orders of Lieutenant T. G. Shortland, at 3 a.m. attacked the 14-gun-brig Adventurer, having a crew of 79 men, lying in the harbor of Corigou, Department of Finistere, France, near the Isle of Bas, in the English Channel. She was gallantly boarded, and carried after a spirited defence. The loss on the part of the British was one man killed, and Mr. Frost, midshipman, and three men wounded. The forts ashore had just opened fire upon the brig when the wind veered round to the north-west, and blew a fresh gale directly into the narrow entrance to the harbor. After being exposed to a smart fire for more than two hours, Lieutenant Shortland, by great exertion, succeeded in bringing out his prize without further loss. For his gallant conduct on this occasion Lieutenant Shortland was made commander. The Junon, of which the discoverer of the Hunter was commander when he received the wound which caused his death, was originally a French 40-gun frigate, and was captured after a desperate fight, February 10, 1809; during which she lost 130 in killed and wounded, including her gallant commander Captain Rousseau, out of a crew of 323. The British frigate Horatio had 7 killed, and the captain and 25 men wounded; the gun-sloop Driver 6 wounded; and the frigate Latona one wounded...Evening News. 29 March 1902
7). 1789 - A chart of the track of the Alexander on her homeward passage from Port Jackson on the eastn. coast of New South Wales to Batavia; performed in the year 1788 under the direction of Lieut. John Shortland - Map maker Thomas George Shortland - Antique Mapart Australia
8). Colonial pardon for James Freeman in exchange for him becoming the public executioner. .March 1788 (State Records of New South Wales)
 HRNSW, Vol. 1, part 2, p. 43.
 Freeman's Journal 24 July 1787
 Freeman's Journal 3 June 1787)
 HRNSW, Vol. 1 part 2., p.78
 Newcastle Morning Herald 13 August 1897