A lot of that depends on if you're using your own computer or the roommate's. Using their PC inevitably involves compromises. For instance, they'd have every right to object if you try to install software. Even then, you can still protect your secrets by using your browser's privacy mode, where the browser keeps no record of the sites you visit or the pages you see.
But you still may not be safe. If the owner of the PC has installed child protection software, they'll be able to follow everything you do on the computer, and block some of it. While I recommend such software to parents, it seems extreme for controlling an adult.
For true security, you want your own computer. Use a strong logon password if you're worried about your roommate getting onto your PC. You still might want to browse in privacy mode, especially if you're doing something that's potentially embarrassing, as well as encrypt any sensitive files on your hard drive. I recommend TrueCrypt for encryption.
But even if it's your PC, what of this person controls the network? Can they spy on your Internet activity there?
It's theoretically possible to track usage through a router's logs, but with consumer (as opposed to business) routers, it's difficult, time consuming, and may not yield useful information. Router logs give you IP addresses, not URLs, and pasting them into a browser seldom brings you to an actual web page. You can look IP addresses up on something like the RIPE Database Search, but the information returned isn't always helpful. To make matters more complicated, the log doesn't differentiate from a connection to a web page and a connection to an ad on that web page. That's a lot of IP addresses to check.
I tried to track surfing records on my D-Link DIR-655 router. I even got a D-Link technician to help me. We came to the conclusion that someone who really knows what they're doing, and is willing to spend a lot of time doing it, might get some useful information, but probably not.
If you really believe your relative or roommate would go to such lengths to spy on you, you have three options: Have a long and frank discussion, buy your own PC, or move out. I'd probably do all three.