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Upgrading may be a possibility but my approach here might be to spend some time cleaning the computer, both physically and softwarely. To clean it physically, first unplug computer from the wall, open the case and blow out the inside with compressed air paying particular attention to the heatsinks and fans. Be careful when cleaning it physically.
To clean it 'softwarely', this guide should get you far in increasing the speed back to how it was when new http://www.tomshardware.com/forum/8263-63-simple-free-guide-removing-malware
Something also to consider is that the hard drive may be getting full (more than 70% capacity) which can slow down your system a bit, defragging the hard drive may also help. Another possibly is adding an additional hard drive - better yet a small SSD (say 128GB or less) to which your operating system can be copied to may bring you some speed without any further changes - like this one which comes with data copying software (I would personally only transfer the operating system to the SSD or use it as a cache drive) Crucial M4 2.5" 64GB SATA III
for $82. Note that some shopping around may yield better prices.