William Hamilton was entered in the Navy List of Medical Officers in 1814.
He was employed as Surgeon-Superintendent on three convict ship voyages to Australia:
Elizabeth to New South Wales in 1818. Maria to Van Diemen's Land in 1820 Norfolk to New South Wales in 1825
William Hamilton received his appointment as Surgeon Superintendent on the Elizabeth on 30 May 1818. He kept a Medical Journal from 9 July to 25 November 1818. The Elizabeth departed Cork on 26 July 1818 with 101 female prisoners and arrived in Port Jackson on 19 November 1818. There were no deaths on the voyage.
William Hamilton returned to England in April 1819 on the Shipley with seven other naval surgeons 
The Maria departed England on 10th August, came direct and arrived in Hobart on Friday 1 December 1820 with 156 male prisoners.
After this voyage William Hamilton applied for a land grant, however it wasn't issued immediately despite high recommendations:
Extract of a letter from Lieut. Governor Sorell, dated Hobart Town, 12th Decemr., 1820, addressed to Govr. Macquarie.
Mr. Hamilton, Surgeon of the Maria, is I believe known to Your Excellency, having been before in New South Wales.
He has brought out his Convicts in a manner highly creditable on this occasion.
This Gentleman and Captain Walker of the Maria have stated to me their intention to become Settlers in this Colony; but, being necessarily obliged to return home, they have expressed their wish to have some reserve of land promised to them. From the usage in similar cases, I felt more than doubtful of the admissibility of the application; but I considered it as bearing affinity to a recent instance, vizt. that of Captain Taylor of the Ship Caroline, who has delivered to me an order from Lord Bathurst for a Grant of Land in this Settlement subject to no conditions.
I, therefore, thought it right to lay the application of Mr. Hamilton and Captain Walker before Your Excellency, under the impression that both, and particularly Mr. Hamilton, would form a real acquisition to a new Colony. I explained to them clearly that I could not take upon me to make any engagement of reserve in Land; but I assured them that I would state the matter to Your Excellency.
Certificate from Lieut. Governor Sorell.
Certificate for Van Diemen's Land.
I do hereby certify that Two Officers, Thirty two non- commissioned officers and privates, as the Guard, with Five Women and one child and one passenger, and one Hundred and fifty Six Male Convicts were landed on the sixth Instant from the Ship Maria, Mr. Hamilton, Surgeon Superintendant ; that the Convicts stated themselves to have been treated with humanity during the Voyage, and that they landed in a state of health, order and cleanliness, highly satisfactory to me and creditable to the Officer in charge. I further certify that I have examined the Journal of the Surgeon Superintendant, and consider his management of the Convicts to demand the fullest testimonial of approbation. Given under my hand at Government House, Hobart Town, this Twelfth day of December, 1820. Wm. Sorell, Lt.-Govr.
Application for Land Grant
On the eve of his departure as Surgeon on the Norfolk in 1825, William Hamilton made another application for a land grant:
10 June. Surgeon Hamilton to Earl Bathurst.
Norfolk Convict Ship, Spithead, 9th ApL, 1825.
I have the honor to be a Surgeon in His Majesty's Royal Navy ; and I am at present employed (my third Voyage in this Service) as Surgeon and Superintendant of this Ship, taking 180 Male Convicts to New South Wales, and, as I will at the end of the Voyage be entitled to a retirement of Fifteen Shillings a day, I am desirous of having a Grant of Land given to me in the Colony, or in Van Dieman's Land, which ever I may prefer; and I trust that upwards of Twenty nine years of Actual Service will give me some claim on the Indulgence of His Majesty's Government.
I have therefore to request that your Lordship will be pleased to give such Orders, or Directions, to His Excellency the Governor of New South Wales, or His Honor the Lieutenant Governor of Van Dieman's Land, as you may think fit in the business. And having had this Settlement in View before, but which my en- gagement with the Board, I was serving under, prevented my putting in Execution, I beg leave to Enclose for your Lordship's inspection, some Correspondence that took place on the Subject, and I trust the Testimonials of my Conduct therein will Entitle me to your Lordship's Consideration without troubling your Lordship with references for Characters from the Boards I have so long and so constantly served under; as well as to several Officers of the Highest distinction in the Navy whom I have served under.
I have, etc, Wm. Hamilton, Surgeon, R.N., and Surgeon and Superintendant of the Norfolk Convict Ship.
The Norfolk departed Portsmouth on 17 April 1825 and arrived in Port Jackson on 18 August 1825 with 178 male prisoners. William Hamilton kept a Medical Journal from 11 March 1825 to 23 August 1825. He remarked that he found it a difficult aspect of the voyage to deal with men not only so little disposed to assist each other but also to look after themselves.
Two prisoners died on the voyage out.
 Historical Records of Australia, Series 1, Volume XI, January 1823 - November 1825, pp.640-41
 Medical Journal of William Hamilton. Ancestry.com. UK, Royal Navy Medical Journals, 1817-1857. The National Archives. Kew, Richmond, Surrey.
 New South Wales, Australia, Colonial Secretary's Papers, 1788-1825. Copies of Letters sent within the Colony (Ancestry)