Definitely as
@Phaaze88 said. I'd suggest removing the heatsink and cleaning both it and the chip with Q-tips and rubbing alcohol not lower than 90% otherwise it won't evaporate as well.
What I typically do is remove the heatsink, and if the thermal paste still appears a little wet, use a few q-tips, run and spin it slowly and gently over the top of the CPU and heatsink, and when it looks clean enough, then go over it again by wetting another q-tip in rubbing alcohol and running that over both the CPU and the heatsink, letting it dry before you apply any thermal paste, all while changing the q-tip often.
I typically use Antec Formula 7 Nano-Diamond Thermal Compound personally, but the choice is yours. The thing with this particular one I use is that diamonds are Really good for transferring heat, and the paste is non-conductive, so it's safe if it accidentally leaks a bit.
When applying the paste, there are Many different methods out there and it all depends on your particular CPU shape, and heatsink type, as well as personal preference.
Most will do the small pea-sized amount of thermal paste, and that's all fine and good, but some heatsinks don't do so well with that shape, and can't spread it well enough, especially the older socket heatsinks like socket A for example because of the odd way the heatsink has to be mounted which isn't perfectly flat when you first start attaching it. Modern heatsinks are a bit easier but still vary.
What I personally do, is if the CPU or GPU chip has no integrated heatsink,(That large silver metal piece over the chip.) then I'll apply a small amount of thermal paste to the chip, and then wrap my right index finger in scotch tape, being careful not to touch the underside of it where I'll be using it. I then press my finger down gently onto the paste on the chip and slowly spread it around, evening it out, I then throw away the tape and apply the cooler. If your cooler has 4 screws, or 4 push pins, do as
@Phaaze88 said and tighten them in an X formation. Being sure to screw the screws a little at a time on each one until they're all tightened properly.
(X formation being like, tighten the top left corner, then bottom right corner, then either the bottom left or top right corner next, etc.)
Note that many people will say against what I have with spreading the thermal paste, but let's also take note that due to just how thick the Antec paste is, it sometimes needs a bit more help. This is what I've been doing for years now. For chips with an integrated heatsink, depending on the heatsink, I'll sometimes just put a pea-sized amount in the center and let the heatsink weigh it down and spread it, or other times, an X shape with little dots between the X arms.
Hope this helps.
😉
P.S. I'd also recommend looking up different thermal paste applying methods and their results yourself as well to find something that you like the most personally.
P.P.S. Antec Formula 7 requires no setting time, unlike most pastes.