It is perhaps not strictly accurate to say warrant officers were appointed by the Board of Admiralty. In general, they actually received their warrants from the Navy Board, which was the Royal Navy's administrative body until it was merged into the Admiralty in 1832. The Navy Board kept records of candidates for a warrant. When a vacancy opens up on a ship, it would (in theory) be filled by promoting suitably qualified mates with the appropriate specialty, more or less on the basis of experience.
Captains were generally given commissioned officers by the Admiralty, and warrant officers by the Navy Board. Only the petty officers were under the Captain's authority. An exception was when the ship is away from Britain. Then, the captain was allowed to nominate a certified person to the position, but these were subject to re-examination and confirmation by the Navy Board later.
Wardroom Warrant Officers were appointed to the time a ship was in service, similar to commissioned officers. They had the privilege of access to the wardroom, where they dine with the commissioned officers.
The Master was the senior warrant officer on any ship, appointed by the Navy Board for each voyage. To qualify for the position, master candidates were examined by Trinity House for fitness in controlling specific ratings of ships. On this basis they were then appointed to take charge of an appropriate ship's navigation.
Both the Purser and the Surgeon were also appointed by the Navy Board. The Surgeon was originally examined by the Worshipful Company of Barber-Surgeons, and after 1745, the Company of Surgeons. The Purser was more of a businessman, and at least some bought their positions in order to profit from selling subpar provisions.
The Chaplains were originally nominated by individual captains on an ad hoc basis. In 1665, as an attempt at reforming, the Archbishop of Canterbury was made responsible for appointing chaplains. After 1677 this role was assumed by the Bishop of London. Nevertheless, there were usually too few chaplains to go around the whole navy, and so candidates may request postings to specific ships. Their status sort of depends on how the captain treated them.
Standing Warrant Officers differ from the rest by being appointed to a specific ship, rather than specific voyages à la commissioned officers. Even when the ship became laid up, they remain attached as a maintenance crew. Promotions come in the form of being swapped to a larger ship.
The Gunner and the Boatswain were warranted by the Navy Board. The Gunner had to be certified by veteran gunners of the navy as well as in mathematics in order to receive a warrant.
The Cook was typically a disabled serviceman appointed by the Victualling Board, a reward of sorts for his service. This organisation which was responsible for provisioning the Royal Navy ships with food, drinks, and other supplies. It was merged into the Admiralty alongside the Navy Board in 1832.
The Armourer and Gunsmith were warranted by the Board of Ordinance, an independent department that supplied artillery and munition needs of both the Army and the Royal Navy until 1855. These were not usually found on all but the biggest ships.
The Master-at-Arms and the Schoolmaster were appointed by the Admiralty itself. These were petty warrant officers.