Here is a guide for all the legendary ships:
HMS Prince (Navassa Zone)
This ship is similar to the Man'O War ship but with heavier hits and defense. It will try to get away from the Jackdaw with its superior speed while using Mortar on you. Similar to other mortar attacks, you will see the area of effect, giving you a small opportunity to avoid damage.
Try to move close and use the chain and ram strategy, trying to stay as close as possible so that it prevents the HMS prince from using Mortar. The HMS Prince is probably the easiest of the Legendaries.
La Dama Negra (Serranilla Zone)
The La Dama Negra is very similar to the HMS Prince in terms of its attack style with one key difference being that unlike other mortar attacks, it does not show the area of effect.
One way to respond to the mortar attacks is try not to move towards it directly from a distance, as it will most likely fire devastating mortar shots in your path. If you are circling the La Dama, slow down a little when you see mortars coming, as most of its attacks are adjusted for where you are speeding towards.
The best strategy against this ship is to stay close and use heavy cannons while ramming where you can, staying close will also mean reduced chance of it using mortar on the Jackdaw. Remember that her sides are heavily armoured. Moving behind her will allow you to do normal damage.
HMS Fearless And Royal Sovereign (Eleuthera Zone)
This legendary battle is a fight against two with less HP than the rest of the Legendaries, they will come at you by funnelling you between them and using their side cannons against you.
Best strategy is to chain shoot one and then ram them, once the boat is pinned down continue to use chain shot and ram, making sure the boat doesn't get too far away from you. Another crucial part of the battle is to use brace when your boat is between the two.
If you destroy one boat before the other, the remaining will employ a different strategy where it will constantly ram you for massive damage, so another strategy is once you've damaged one ship's health to about 10%, start using the chain and ram strategy on the other ship so that once you destroy one ship, you can finish the other one with one or two mortar shots.
El Impoluto (Dry Tortuga)
In terms of difficulty the El Impoluto is probably the hardest of the lot as it is a lot more aggressive and has massive ramming power compared to the other Legendaries.
Strategy against it is to shoot chain and then ram where you can, however, given its superior movability compared to the Jackdaw, you will also have to employ cannons and mortars when El Impoluto runs away from you, it will respond with cannons, so remember to use brace when you see cannons coming.
Another thing to watch out for is when it moves behind you, the best defence to deter it from ramming you is to let off a few fire barrels and let off a mortar to steer it away.
Source: IGN Assassin's Creed IV: Black Flag Wiki, "Legendary Ships" article
Best Answer
"If you manage to fight through all the Legendaries, apart from the large monetary rewards (don't forget to loot the packages for the money); finishing all four battles will also give the Jackdaw the additional move of a speed ram, where aiming forward and pressing X (PS) and A (Xbox) will allow the Jackdaw to dash for a moment for a raw with increased power."
Here is the complete guide for them: http://m.ign.com/wikis/assassins-creed-4-black-flag/Legendary_Ships