Like Charper said, make sure you have one (at least) case fan set up to pull air into case and one to push air out. The more air flow the better (case fans have arrows that point in the direction of air flow, if you go and buy them, just make sure one fan arrow points in, and the arrow on the other case fan points out of your case after you install them). (note: most power supplies do have a fan, but this fan is really meant to cool power-supply, not enough to pull air through case thereby cooling components and cool the power-supply too.
Another thing about case fans, you can filter the air that comes into the computer, because air is more likely to flow where case fan is, and if case fan on the front is pulling air in, and you put yourself a sponge filter in front of it, it will collect dust/lent, leaving inside of your computer fairly clean. Would just have to clean/check filter often.
Leave your case off, and make sure your CPU fan is working when computer is on (sometimes they quit) so that you can rule out CPU fan. Also, dust is more likely to accumulate if computer is sitting on the floor, especially carpet. If your computer sits on the floor, try putting it up on a table or something, to get it off of the floor. That should reduce the amount of dust it collects. If not on the floor, then make sure you don't have a lot of lent/dirt around computer (say 5 foot radius).
Clips on heat-sinks are different from one another (similar, but most are a little different). Just study where and how clip is positioned, then when you go to reinstall, you'll have a good idea. Better yet, take a digital photo (or 2 or 3) so you can visually check your reinstall of heat-sink. Most of the time they are spring clips, pushing in one spot causes another to raise/lower so that clip springs open and you are able to remove it. Just remove one side at a time, if it is a socket processor, it should have one side that has a spot that looks like it is meant to be pushed on, push down, and it should lower it below plastic that is holding it. After it is down, might have to pull out on it a little, but it should release. Once one side is released, pushing that side towards the side that is still attached should slide it out, enough for that side to become unclipped. Slot processor's are similar, just might have to study a little longer before removing.
If it just taking case off, and cleaning inside of computer and removing heat-sink/fan and cleaning that, it really isn't that difficult. Just don't forget to open your CD-Rom drives and clean them too, open them, and blow air through them, while case is open. So that dust gets out of them. Clean around front of computer where CD-ROM/DVD-ROM drive(s) are at, to be sure to get dust out of there too.
Turn off computer and unplug before trying to clean it (open CD-ROM's before shutting down computer) don't use metal vacuums (just use canned air from the store, it is ok to use Vacuum on the outside of computer, but don't use it inside to be on the safe side). Touch your hand to metal parts of case to discharge any static electricity (note: this isn't the "preferred way" but better than nothing) before removing heat-sink, you'll be close to one of the most sensitive to static discharge components in your computer (second to Memory) having just a little (less than you can feel actually) static discharge can mess up your CPU (or memory or the various chips that lie on your motherboard).
Beyond that, it really isn't hard. Clean like you would anything else (just don't use liquids of any kind). I mentioned static concerns so you would be aware, however, most never have a issue with this, just be aware, but don't be too overly concerned. Also, if you can, you should be able to clean heat-sink/fan without removing it. If you are too afraid to remove it, clean it still on the MB. Just have to make sure you go at it in every direction you can, so that you clean it well.
