In order to install a 64 bit version of Windows, you need to have a 64 bit processor and a motherboard that supports more than 4gb of ram (well, that's not exactly necessary, but why else would you want to install 64 bit at this point in time?). If Windows 7 won't support your graphics card, replace it. If it's onboard, install a separate card then DISABLE onboard graphics in the BIOS to avoid any problems.
I believe Microsoft has a Windows 7 Upgrade Adviser just like they did for Windows Vista. I suggest you run it in order to get specific information about your hardware and potential problems you might encounter if you upgrade. If your computer passes, I suggest you download a pirated version of Windows 7 (NOT the Beta!) and install it just to be sure it works properly with your machine. It'd really suck to have MS tell you your computer will work fine with Windows 7 and you go spend $300 on it and then after it installs you find things just don't work right. The Upgrade Adviser tool isn't exactly the most sophisticated program in the world. Think of it as a program like MS Paint.