With the 1.4.5 patch now on the PTS, I was able to hop on and check things out. Specifically, I took a look at how to claim a Fortress. The primary prerequisite is that you need to be an Alliance Leader in order to make the claim. Thankfully, I was able to get an Alliance setup to test with. On to the Fort!
Getting to the Fort
In every tier 4 warcamp, there is a new flag which will teleport you to that pairings fortress. It is only active if your fort still has a relic. If it has been taken, the flag will not work and also look very drab.
Claiming the Fort
Once you are at the Fortress, head to the 3rd (I think forts have 4) floor. There you will find an Alliance Altar. Click on it, and a few OK's later you are now the owner of the fortress. Your guild news window will mention the fort was claimed.
Money in the bank
Owning a fortress is not free. It costs 24 gold per hour. The gold is taken from your guild bank, so make sure you have some money in there. The Relic guardian, located on the top floor by the relic, also acts as a Guild Banker for members of the controlling alliance.
Once you claim a fortress, your Guild Banner will be used to decorate it in certain places. this is a nice touch from keep claiming which has returned.
That's all there is to claiming a fortress.
The main problem with the system on the PTS is there really is no feedback. Looking at the banners seems to be the only way to tell if the fortress is claimed and by who. The map does not provide any tooltips like keeps used to give.
The other problem is that bonuses received by being the owner of a fortress do not show up anywhere. Hopefully Mythic can put in some kind of indicator (buff icon) that lets you know that you are receiving the bonus.