I’m assuming that this is a Windows computer.
First step- make sure the hard disk is defragmented and checked for errors.
Next step- make sure that your computer is not running unnecessary programs in the background. I recommend Process Explorer from SysInternals as it’s much better than the built-in tool. There are an array of startup managers available as well if you don’t want to go fooling about in your Registry (and why would you?) so that once you have only the programs running that you want to run, you can make sure they don’t start up again on the next boot.
For the next step, make sure your antivirus has a mode that lets it run at startup, before the OS loads and boots. Several antivirus programs offer this capability, I will leave it to others to recommend specific ones.
Now reboot and let the full virus scan run. Time to go for lunch. When you get back, if the antivirus scan has found and eliminated viruses then your computer will probably run faster.
If not, then the next step is to re-install Windows.
Note that if you ran Linux you would never have to do any of the above.
Note also that the real first step is to make sure you have a current backup. You make regular backups, right?
If the above instructions don’t speed up your computer the next step is to get a bootable CD like PartedMagic or Ubuntu 10.04. It needs to be one that offers MemTest86+ (most all of them do.) Reboot again and run MemTest from the prompt. This test will take a long time unless you have a small amount of RAM, and if that’s the case then the answer is “add more RAM and your computer will go faster”. Any amount of RAM less than “enough” will slow down your computer a lot. RAM is cheap these days. Max it out.
If MemTest finds errors, replace the RAM (the program will tell you which bank) and the computer should run faster.
If it doesn’t the next step is to check out the hard disk. If you are using the Ubuntu livecd you can use an included program called Palimpsest to run tests on your disks. Run a “SMART Test”. Do the full one. It may take several minutes, but should be quicker than the virus scan was, so go to coffee instead of lunch. If the test finds errors you may need to replace the drive. Make a backup first, of course.
Once the drive is replaced then your computer should be running fast again. If these steps don’t work, call a priest and have the machine exorcised. Beyond that I don’t know what else to tell you.