How can I find out what IP address was used to make purchases from my account?

The thing is I'm 100% sure it was my cousin as he recently used my account for a night and this happened the next day. He completely denies the fact and I can't prove it without any hard evidence, that's why I need this information.

There would be no way for you to find that out. If you believe your account was compromised, you would need to contact support and have them start an unauthorized access investigation.
support.xbox.com/contact-us

Greetings!

I assume your university's network has a set of public IP addresses that can be addressed to the Internet-at-large. Behind the WAN interface, are the local interface, facing the internal, or private side of the university network. These addresses can be
used to share multiple host machines with IP addresses that are not routable on the Internet.

Access the web utility interface of your router and make sure the WAN, or Internet IP address are not an address within the private range. If it is, then you are behind multiple layers of NAT.

Let me know what you are able to find out.

I have figured it out. In the LAN between laptop/xbox, the laptop was given IP 192.168.0.1. I then had to tell the xbox that the DNS should be the laptop's IP, 192.168.0.1. Now it works fine, but I'm trying to find out how to use my account here with my
profile on the xbox. When I enter the email address it says that the account is accossiated with another profile, but it isn't..

Still, do you think you can find out the model number of your 2wire device. This would help us a lot.

Could you do a simple check and make sure that the WAN/Internet IP address that your router receives from the bridged connection is not an IP address within the private range. Only post up the first two octets of the WAN IP address and not the entire address
(e.g. 192.168.xxx.xxx).

TIA!