How to fix a leaking cold/hot water faucet
Step one: purchase the appropriate valve for your sink.

Step two: read the instructions that come with the valve. :)

Sorry, I couldn't resist.  Turn the water off to the sink first.  The faucet may have a cap that can be removed, either via a small screw in a difficult to see place (behind or when you lift the handle all open) or the cap is friction fit you can pop it off with a putty knife.  My best guess looking at this valve and because it's for separate hot/cold handles is that the cap is on the very top of the handle.  From there, you can remove the handle, usually with a screw or two.  Unscrewed, it should come right off.

And then the cartridge/valve is removed by unscrewing it.  However, this is the tricky part, and the one that took me the longest time to figure out.  At least until I got the replacement valve that included an adapter to unscrew the old and install the new valve (buying the valve first would have saved me an hour).

Before you install the new one, check if the instructions tell you to lubricate the o-rings.  If they don't specify what kind of lubricant, I think Vaseline is safe, though I've been known to use some chapstick in a pinch.  You should also clean out the hole in the sink as best you can, using an old toothbrush or whatever else you can get in there and not lose.  After that, you just reverse the steps you took to get everything apart.