If I bought the Iphone 5 at the apple store am I under a contract?

When the Iphone 5 came out I bought the phone at the apple store then after that I went to AT&T to get the phone set up. What I'm asking is am I under a contract with AT&T or do I own the phone since I bought it from apple. I want to upgrade to the new iphone 5s when it comes out.

Depends on if you paid retail for it or not. If you paid retail (probably more than $200) then you are probably more than likely not under contract. If you paid around or less than $200 then you are definitely under contract. Why don't you just call your carrier and ask them? Better than wasting time on here.

Any service like AT&T has a contract plan for 2 years so I would check your AT&T account you can do that online. To see if you are eligible or just pay for the 5s at full price

No you are not under contract. Contract is only for iPhone 5s and 5c and not with iPhone 5

You own the iPhone.AT&T has no claim to it since you did not get a discount on the iPhone from them.
HOWEVER when you signed up with AT&T you may still be under a service contract with them that is not connected to the iPhone.
When you "upgrade" you do not have to go through AT&T either. Simply sell your iPhone back to Apple or a private party and use the money for the 5S.

NOTE: Wireless carriers offer handsets (phones) to their customers at discounted rates with the understanding that they will earn the difference back through the monthly service fees paid by the customer. If the customer breaks the contract the wireless carrier is out the money it spent to buy the phone initially and will then put a lien on the phone. This is how "bad ESN" come about. That phone can no longer be activated on their network.