Adapted from these two older answers:

As a general rule, no, there were not "hard barriers" of wealth or bloodline, though like in most jobs wealth and pedigree wouldn't hurt. The most famous admiral of the age was the sixth child of a Norfolk parson.

During the 16th century in general, naval ranks became more specialized. While navies of the Armada, for example, often mixed command among soldiers and sailors with the "master" responsible for navigation regardless of who commanded, the naval ranks were becoming more regularized. The most obvious example of this was a sharp line between commissioned and warrant officers. Warrant officers were those men such as the boatswain, the surgeon, the carpenter, the gunner, the purser and their attendant mates, as well as the master, who as the WO in charge of navigation and sailing had a somewhat exalted role among the warrant officers. The status of warrant officers got a boost in 1787 when they were granted official uniforms, but there was always a divide among warrant officers who were essentially gentlemen at sea (surgeons, pursers, chaplains) versus boatswains, gunners, carpenters and others who were usually former common seamen.

Starting roughly in the 1760s, the way in which men became officers in the Navy became more regularized than it had been before. In general, there had always been a boom-and-bust cycle in manning the commissioned ranks, in which large numbers of new officers were commissioned during wartime only to be thrown on the beach in peacetime. This led to absurdities like 40-year-old masters' mates (midshipmen) serving with 12- and 13-year-olds, but there was no good solution to this when seniority was the main way of gaining promotion. Once a man had made post-captain, he was essentially assured of eventually becoming an admiral, barring death or gross misconduct, but officers before reaching that step would not have any guarantee of promotion. Many of these disappointed officers served in the merchant navy, East India company, or other navies (navies of other powers). The creation of the navy's Transport Service also offered some honorable employment for officers who had a great deal of sea time, with a moderate hope of promotion.

The "discovery of the longitude" in the 1760s meant that mathematics became a bigger part of a sea officer's education. The invention of the chronometer offered a simple and reliable way for officers to tell the ship's latitude (by its deviation at local noon from Greenwich time), but very few ships had Admiralty-supplied chronometers until the 1840s. That meant that lunar observations were the most common way of finding out a ship's east-west position in the ocean, and those required a finer understanding of mathematics than had been required before. The need for mathematics also coincided with the fact that many boys who went to sea had little formal schooling to lead to a process in which boys going to sea would often spend some fictional "sea time" on the books of a ship while actually at a primary school on land. This was theoretically a crime, as the ship's purser would be paid for the rations drawn by the nonexistent people on the ship's books, but it was understood as a necessary minor offense to allow boys to get some education other than that at sea.

So the typical future sea-officer might be entered on a ship's books at 12 or 13 while getting some education on land and not join the ship until 15 or so, while they would "learn the ropes" in a professional manner. The nascent officer would take an examination to become a lieutenant, in which he had to offer journals of his sea-service and answer questions from an examining board, where he would be grilled on practical seamanship, navigation, mathematics, geography, history or really anything else the board wanted to throw at him. There wasn't a standard examination, so it was quite possible that sympathetic boards could pass men rather easily (Horatio Nelson's stepson rather famously became a lieutenant at around 16 and a post-captain at 17, due to Nelson's intervention and "interest" in his future, despite being spectacularly unqualified for the job).So the first barrier to entry from a midshipman to a lieutenant was sea-time and passing that examination.

Once a man became a lieutenant, he would have to rely on luck and his own "interest" from sponsors for further promotion. The first lieutenant of a ship was often promoted after a successful action (this was seen as a compliment to his captain, oddly enough). A lieutenant's next step would be as a master and commander of a vessel; he would by courtesy be called a captain at that point, but it wasn't until he was promoted to captain of a post-ship that he would be called a post-captain, or just captain, and be assured of (if he lived long enough) dying an admiral.

Once a man was a post-captain, his further promotion was entirely based on seniority on the list of captains. (What's a post-ship? In confusing logic, a post-ship is what a post-captain would command. In practical terms, it usually meant at least a frigate, although sometimes smaller ships were considered post-ships out of courtesy to a captain.)

Now, while that deals strictly with promotion, the related question is what kind of assignments a man might encounter throughout his career. By the time of the Napoleonic wars, seniority (while still ruling when a man might become an admiral) was not the only way for men to reach the upper ranks of the Navy. In 1743 George II had agreed to expand the number of admirals from 10 to 21: there could now be more than one in each rank. Before this, admirals were by convention divided into red, white and blue squadron, commanded by an Admiral of the Fleet, each seen as a distinct rank and promotion only by seniority.

By 1747, though, the Navy had one Admiral of the Fleet, six admirals, six vice-admirals and eight rear-admirals. Even with that expansion in rank, there weren't enough admirals for all commands, and there was still no way to promote men to admiral from the captain's list; seniority was the only step. So by 1747 a rank of admiral "without distinction of colours" was created for the specific purpose of superannuating men into a non-active rank, which allowed the Navy to reach further into the captains' ranks to promote men to admiral (though with the implicit tradeoff of retiring from service anyone senior to them).

One of the ways to attract the attention of the Admiralty, besides having connections with people in government, was to be an active and enterprising captain. Captains of frigates and smaller ships that cruised against enemy shipping always had opportunities to endear themselves to their betters (not least because prize money was shared with an admiral whose orders a captain was sailing under), but enterprise in general was generally valued by the admiralty. Lieutenants and commanders who had washed up "on the beach" without an assignment sometimes volunteered for Admiralty contract work, such as surveying or volunteering on hired ships, to catch the eyes of those above them; unemployed captains could resort to privateering (though this was looked down upon socially) or letter writing campaigns. But for many captains, assignments and promotion could be based on luck as much as skill or "merit" broadly defined.

A good example might be that of Philip Broke, captain of HMS Shannon made most famous for its brief but bloody fight with USS Chesapeake in the War of 1812. Broke had commanded Shannon since May of 1806, and had been active in various naval expeditions -- bombarding coastal towns, surveying the polar north, escorting transports, that kind of thing -- and had captured a small French privateer in 1809. Broke was interested in naval gunnery, and had drilled his crew to a high standard both in terms of aimed fire and rapid fire, but had not been singled out for any particular promotion or distinction as a result of that until the War of 1812, when he took command of a squadron blockading American ports. Through a series of events, Shannon was alone off Boston Harbor in the late spring of 1813, and Broke decided to challenge the American captain James Lawrence with USS Chesapeake to single combat. Chesapeake never received the challenge, but sailed out to meet Shannon on 1 June; after an intense exchange of gunfire that lasted about 12 minutes, Chesapeake surrendered to Broke, suffering 48 killed and about 100 wounded.

Broke was himself wounded during the fight, taking a cutlass blow to the head that incapacitated him. He never again served at sea, but was created a baronet, became a training and gunnery expert, and died a rear admiral of the red squadron. All of his experience and work led up to the fight with Chesapeake, but without the "luck" of them coming out to fight, he may not have had the choice to prove himself particularly noteworthy in the RN.

Does that answer your question? Let me know if there's more I can add.