Understand if you need a new Computer
The easiest way to understand if you need a new computer is this - You need a fixed frame rate, it has to be at least 60-59fps, if it drops below this you may have problems with aiming in crucial situations and you'll have "lags", best would be always above 60fps, if your computer can't stand it, you may consider buying a gaming computer.
Other than fps?
You'll surely perform well with a better mouse, a 144Hz monitor, a more powerful CPU... but if you can't spend money for such things, as long as you have 60fps you're good.
From here on is a list of good equipment for your computer, from mouses to headsets, for every budget!
The order is casual, it's not a ranking from best to worst, they are all good.