Clicks and pops are the most irritating thing about playing records. Cleaning them is a good way to reduce the number of clicks and pops, but some will still get through.
There are a lot of different types of de-clicking software out there. They vary hugely in cost and effectiveness, not all the expensive ones are worth the money. Nearly all of them will change the sound of the file as they remove clicks. The trade-off is to find one that works well enough for you without mesing up the sound of your LPs. Common sorts of 'artifacts' you may hear from de-clicking a record include
I now use Wave Corrector if I want to de-click records. The results are pretty good and it seems to cause less distortion than the (old) CoolEdit tool. There is a good tutorial on using Wave Corrector here.
I used to use an old version of CoolEdit Pro after having tried a few other programs. The new version of CoolEdit does not include the click and pop filter, so you will have to buy it seperately. Have a look at the resources page for alternative programs. Below is a short guide to declicking with the old version of CoolEdit Pro.

You do not even have to declick a recording. You could just filter out the click that you can hear (very time consuming but can produce good results). If the clicks and pops do not bother you then leave it as it is and skip to the next section.
To declick choose the Click/ Pop Eliminator on the Noise Reduction section of the Transform menu.

CoolEdit can find the levels automatically for you. Choose 'Find Threshold Levels' The new version of CoolEdit will set ALL these settings for you if you choose 'Find ALL Levels'. Use 2 or three passes to help mop up any stray clicks that may have been missed.
When the settings are adjusted to your recording click on OK to start the process.

Processing may take some time, but in the end it is likely to be worth it. Below there is a screen shot of the results of the very noisy recording shown above. The remaining clicks are relatively easy to spot, and you can zoom in on them an filter out the high frequencies.