The clips I assume is either for a different type of cooler, or they are there to make sure the whole thing really sits tight.
The fan however is usually fastened to the heatsink using small screws, it depends on the maker of the fan and heatsink really. You may or may not be able to remove the fan.
The cpu itself is stuck to the heatsink with a paste or glue, I would not recommend loosening anything as it could loosen the contact the cpu has with the heatsink which could result in failure to cool properly.
In that case you need to clean the cpu and heatsink and apply new paste or glue and re-seat the whole thing.
The compressed air cans do not cost that much, and they are usually pretty big so you will not only be able to clean your cpu sink/fan,
but all the fans in the PC as well as cd-drom, mainboard and the entire case, and still have plenty left until next time you need to do it several months from now. The can should last enough for 2-4 cleanings of the entire pc (including keyboard

And I'd recommend blowing the pc clean 2-4 times per year, depending on the amount of dust in the room and that the PC collects.
Not only is the dust a fire hazard. (ever looked inside somebody's TV set? Eeek!) But too much dust prevents the heatsink from dissipating heat properly. (the dust acts as a insulator i.e. keeping the heat)
Not to mention the dust can cause fans to malfunction.