There isn't a lot you can do to an air conditioning system without specialized service equipment. Anything other than topping up with a can or two of R134a is best left to the pros..
You can do a cursory inspection by looking at all the lines and components.. If there is a fairly big leak, you will see oil staining the area of the leak. Pro shops will sometimes put an ultraviolet dye in the line. Then they put it under pressure and inspect the system with an ultraviolet lamp. Wherever the system is leaking will glow in the dark when you shine the light on the dye that leaked out.
If your compressor is leaking, you'll need specialized pullers and tools etc to rebuild it. (Although you could replace it with a rebuilt or a used one).
Note that when you replace anything, the system has to be opened to the atmosphere. This lets air and water vapor in.. Before you refill the system, you NEED to put a high vacuum on it for about 1/2 hour to suck out the air and water vapor that got in. You need a vacuum pump for this.
And finally, you have to attach gauges to the high and low pressure ports to make sure you have enough of a charge and there are no obstructions inside the system. There's a good article explaining how to using the gauges at this link:
http://www.aircondition.com/tech/questions/82/