Assuming this is a desktop pc. Dirty fans are not a massive problem. Obviously don't let them so ridiculously dirty to the point they are not spinning. When it comes to computers overheating there are two main things to look at. First and most common is the fan/cooler on the graphics card not spinning due to dust. The next is that the heat sink (NOT THE FAN) on the CPU is covered with a wad of dust. This is very common. Both of these problems usually result in instability of the machine and with the Graphics card (If you have a dedicated one) the display may shutdown randomly and the computer continues to run.
However it does not sound to me like your problem is cooling, again if this is a desktop. I would suggest that the heat from the machine is normal. My first port of call would be the power supply. PCs are getting build more and more with poor quality power supplies and they are almost always the first part to go. Usually without warning but it is not uncommon for them to produce the plastic burning smell your talk of. It is the most likely component to do this and it happens often. If this is the case stop using the computer and have the PSU replaced. I have seen this 3 times in the space of a year. On two I replaced the PSU and the first of the three failed entirely and took out most of the machine at the same time.
Ruffles