2007 xk alternator?
 

So my 07 has been dead several times since I purchased it in November. At first I thought it was because we were leaving our key in the car and not locking it. From there I thought it was a few little things but now I'm not sure. I have a trickle charger and yesterday had the batter fully topped off. I drove it once yesterday after that, parked it last night and locked it. This morning the battery was dead. Last week the same thing happened so I took the batt to Autozone and they said it was still good so I again topped the battery off on the charger. I guess at this point I'm going to check the alternator once the battery is charged. 13.5 is what I should be looking for correct?

Unless the shop did a load test on the battery, you will not know for sure. But electrical issues with the XK are invariably traced to low battery voltage caused by a failing battery, or infrequent use and short trips which do not allow the battery to charge fully. There is virtually never anything wrong with the car ( or with its alternator).
Solution: new battery and a CTEK charging unit which you mount permanently and use *always* when the car is not being driven. The result? Zero problems.
Please believe this. There are numerous threads on this issue and on the easy solution ( which I have just stated).

I didn't check what the 2007 MY charging system is but I believe that with the engine running at idle and you put a meter on the battery you should be seeing around 14.2 volts. Check it with the car running and if it is at that voltage then your alternator is OK. As for the battery you should get a load test . Also if you have the battery charged at 13.5 V and you remove the charger and lock the doors see what the battery voltage looks like in maybe 2-3 hours later. If down below 12.4V then you have a battery problem.

Ok, batt is on the charger so I'll test it at fully charged and then a few hours later. I'm pretty sure it's the battery though. My alternator tested out fine yesterday once I had it fully charged. But now it's to the point where I drive the car once and then the battery is almost dead.

You already stated what you're issue is once you said that your battery "has been dead several times". Even a new battery is diminished when brought down, do that several times like you said and you've serious compromised it at best. Get a new one, preferably an AGM, immediately put it on a trickle charger to bring it up to it's top level (no batteries are not "fully charged when they are new contrary to most peoples belief nor will your alternator bring it up properly) and you'll be fine. Your currently battery is toast