As opposed to the four starting factions, to win with the Vampire Coast you need to gather enough infamy to lure out certain pirate captains to take their three sea shanties and also get a special harpoon and then fight a final battle. Honestly, that makes this campaign a lot easier, since you wont have to do rituals and are not competing with the other main factions for the same ressource.
The harpoon can be gotten early on, it is in the possession of a pirate crew that stays in the center of the ocean. It can be beaten rather easily, even early on. If you want to make sure, use two armies and rush them.
Getting the shanties requires becoming truly infamous. Infamy is gotten by a multitude of methods and CA actually managed to make being a pirate the best way to do it. Here are the best ways to do this:
1. Attack (weak) harbour settlements and establish a Pirate Cove. Then build the Smugglers Cove. Have one pirate army just go around and raid the far off harbours, so you wont have to fear a retaliation.
2. Finish missions, especially those for treasures. For that I recommend having two to three heroes, whose only job is unearthing treasure. I have one in Lustria, one in Naggaroth and one in Nehekhara and could unearth a treasure every two to three turns. You will find enough treasure maps to keep them more or less constantly busy.
3. Beat legendary pirate captains. Most of them will declare war on you anyway. Some are harder to beat in the beginning, but by the mid-to-endgame you should easily kick anyone of them of the plank.
You can spend infamy on loyalty and legendary captains, but they are not necessary. You can get them, but don't feel like you must.
Every so often you will have gained enough infamy to trigger a shanty holder to appear. Those will go after you directly, so make sure you are not to weakened by whatever got you that much infamy, otherwise they just might break you. But if you are prepared you should be able to easily beat them.
Having the shanties and the harpoon allows you to fight the final battle. It will be a defensive one against equal parts Dark Elves and Vampire Coast units. So be prepared for lots of infantry and quite a few monsters. I recommend plenty of artillery, especially the Queen Bess and a few gunners. Have a gunnery wight ready to refill the Queen Bess. Your frontline should be able to hold out, so a Vampire with the Lore of Vampires is helpful. Some flyers could ruin your artillery, so keep something flying in your backhand.
The Dark Elves shouldn't be too hard, but what you need to watch out for are the Bloated corpses. Snipe them any way you can, they can easily ruin this battle for you.
You will get a trickly of reinforcements, use them to plug holes in your line. The Corpses you get you should keep behind and wait for the chance to blast multiple units at once.
Afterwards you have won the campaign.
General hints for the Vampire Coast are as follows:
You can easily be unrivaled in the sea, so use that. Raid far off harbour settlements, establish Coves and leave.
Queen Bess is your friend. With a Gunnery Wight and the Lore of the Deep by her side she will literally destroy units in one hit. Having a solid firing line is the base of every army in any case.
You lack fast and heavy units, so prolonged fights are not our forte. Retreat if necessary and don't fight losing battles.
I think the Dreadfleet is the easiest by far. You can actually win with this one without having to recruit additional armies. Concentrate everything on Count Noctilus, have a Queen Bess in his army, cannons, mortars, a Gunnery Wight and Handgunners. Then research the followers that make those units even stronger like Master Gunner and Powder Monkey . Keep raiding the oceans and harbour areas and you will quickly have a fully build fleet with high level units.
Luthor Harkon should work as well, you should ally with the Skaven soon and sandwich the Lizard Men. Keep them alive, take only a couple of regions for yourself and then harrow the seas with him and maybe an additional legendary captain.
Anyway, good luck and have fun.