How to win with Greyjoy in Game of Thrones board game?
The Greyjoys are a challenging house, and require a really aggressive play style. Out of the 6-player games I've played, the Greyjoys have won the least often, but not by a huge discrepancy, so take hope! There seems to be 2 working strategies, both based on the close relationship the Greyjoys have with the Lannisters.
Lannister Bloodbath
Screw the Lannisters early. With the support ships and larger army you start with, paired with the card that adds +1 to each ship, you're basically guaranteed two early victories over the Lannisters. Use these as best you can, because if you aren't in an advantageous position after that, you've lost. You should move into the Riverlands, stopping the Lannisters from growing early. Keep your ships present to support and harass, which helps stop the Lannisters from recovering. Take the Twins, at this point you should have 3 castles and a good control over the mainland.
Things to Avoid

Harrenhall. The cursed castle is cursed in this game as well. Taking it almost always means you're overstretching, leaves you exposed to Lannister, Stark and Baratheon attacks.
The Starks. They aren't a threat early. Don't waste time and precious momentum moving north, preparing for the Starks, etc.
Relaxing. If you do win the initial push, you need to keep up constant fighting to get your first cards back and finish off the Lannisters with them.

Outcome
After 2 or 3 turns, either you are hopelessly crippled, or the Lannisters are.
Uneasy Alliance
If the thought of essentially being out of the game after turn 2 sounds too risky, strike an alliance with the Lannisters. This is easier to do if the Tyrells are likely to be aggressive, as the Lannisters to not want to be stuck between you two. Give them the riverlands, make sure you keep the twins and your starting castles, and turn your attention North. You have a better claim on whiteshore and those Westernmost castles than the Starks do, use your early strength and sea support to take and keep them. Establish a strong base in the North, then decide whether to take Winterfell, turn to the Eeyrie and center of the Board, or betray the Lannisters.
Things to Avoid

Trusting the Lannisters. Don't attack them early because you need a quiet border, but at some point one of you will break the truce. Accept it and be ready.
Harrenhall. Once again, a cursed castle. Not worth it, it doesn't give the Lannisters anywhere else to go and encourages betrayal. Leaves you open to the Starks, puts you in contact with the Baratheons when you don't really need another front.
Raiding too far south. Sometimes you can encourage the Lannisters to stay faithfull by raiding the Tyrells. But it takes a lot of resources and stretches your navy thin with ultimately very little in it for you. And that sneak attack all the way to Highgarden that nobody expects because ship travel is so OP? Never, ever works.

Outcome
Slower game, expect a slog-fest with the Starks, a lot of back and forth pushing. You probably can't ever knock them out of the game, just contain them. Last few turns will be extremely aggressive and bloody. Pray that the Tyrells and Lannisters stay enemies.