If you use Google to search for for "8051 PID" (without the quotes), you will get over a quarter million results. So your first step should be to do that and start reading.

Implementation of PID is just a bunch of arithmetic. Whether PID on an 8051 will work for you depends to a large extent on how fast the thing you are controlling responds to various stimuli, both from your controller and also from the system's environment. For a system variable that changes slowly (like the temperature of some water in a big tank) the 8051 will have no problem keeping up. For something that changes very quickly, however (like the position of a high speed motor), the 8051 may not be able to perform the calculations fast enough for effective control.

-- Russ