I've written a couple of times previously (here and here) about how promotion worked for officers in the period, and those answers might also be helpful for you to take a look at. To set the table a bit, I'll quote from one of those answers:

To become an officer in the Napoleonic period, a man would need to have entered the navy at a suitable age (certainly no older than 14) to build up enough sea time and practical experience to pass a lieutenant's examination, which required that the examining board be presented with evidence that he had spent six years at sea (at least two of them in the Navy) and that he "appeared" to be 20 years old, which was the minimum age to become a lieutenant. And yes, the point of boys going to sea that young was eventually to attain commissioned rank.

In the period of the Napoleonic wars, many boys went to sea as young as 11 or 12 to start building up the required sea time, or were at least put on ship's books by captains who were friends or relatives of their families to start building up the required sea time. It wasn't uncommon for boys to be placed on the books even earlier, and this practice (though technically illegal) was viewed with some benevolence, because as navigation (particularly finding longitude) became more complicated, most boys needed a few years of schooling ashore, particularly in mathematics.

Once a midshipman had built up the required sea time, he would have to sit for a lieutenant's examination in front of a panel of captains, 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).

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 to get to your actual question, if I recall correctly the topic comes up when Jack considers making his coxswain, Barrett Bonden, a master's mate, which would set him on track for eventual promotion if he were worthy. Bonden declines, but nominates a nephew. If I also recall correctly, in a later book (The Surgeon's Mate, maybe?) Aubrey and Maturin cross paths with Sir William Mitchell, who famously first appeared in Navy records rated as an able seaman in 1766 and then became a master's mate sometime between 1777 and 1781. Mitchell passed for lieutenant in 1781, was made post-captain in 1790 and became an admiral in 1808, though at this point he was on shore service commanding the Sea-Fencibles and was thus a "yellow" admiral, one not selected for service with a squadron. Mitchell died in 1816. There is a story that he was flogged round the fleet for desertion, which Aubrey mentions, but the story only appears well after his death and is sketchy at best.

So that's a long way of saying that while it was quite possible for ordinary sailors to become midshipmen, then pass for lieutenants and so forth, it wasn't exactly something that was common, especially in the time period of the Aubrey-Maturin novels (roughly 1801-1820 or so, with a lot of hypothetical years in there). Something to keep in mind is that the increasing, for lack of a better term, "professionalization" of the navy was ongoing, and the long, long competition between the tarpaulin captain and the gentleman captain was being won decisively by the gentlemen. Of the boys who became master's mates, a very small number would even pass for lieutenant, and fewer of them would be made commander or even captain -- the winnowing effect was so great that men such as Mitchell were very much the exception.

(Oh, and by the way, the British spelling is "lieutenant," they just say "leftenant.")

FYI, it is always spelled 'lieutenant'. The British 'leftenant' pronunciation is based on an older pronunciation of the word, while the French pronunciation changed to 'lewtenant'.