To test caps and relay:
Unplug UNIT! REPEAT, Unplug it!
Remove front cover which usually gives access to electrical components. Look for electrical diagram. Scan it in here if you can - it helps to know which lavor unit you have.
Run capacitors are metal cased. Start caps are always plastic cased. Compressor will have three wires, often but not always black, red, yellow. Run cap will be between red (run) and yellow (start). A MALLER run cap may be present for the fan motor; ignore it as the fan runs okay. Run caps are likely OK unless bulged. Start cap, if even present, will be plastic cased and connected from run to a relay. Relay will either be connected to start and common (black)[potential type] or line and run [current type] depending on type relay. In SOME units the capacitor is mounted right on the large electrical box on the compressor, just as some fan run caps are mounted right on the fan motor. Some units have no run capacitor at all (CSIR), others have run but no start cap (PSC). The no start cap kind often benefit from a "hard start kit" which improves start torque. (Makes them a CSCR system).
Before sticking your fingers in there, use a long screwdriver or jumper wire and discharge any / all caps by shorting them, terminal to terminal, NOT terminal to case. After discharge (those things really BITE!) mark the leads and remove the cap. Take it to a motor shop, HVAC or supply house and ask that they test it. Purchase new for any bad ones. SAME value is best. Relay testing is difficult unless you have the right instrument. Simple tests: Current relay, turn upside down and use ohm meter across the contacts. Should be closed. Right side up, should be open. Replace if in doubt, although they cause little trouble. Potential relay contacts should be closed at rest. Coil test with ohm meter will be several thousand ohms. If contacts are stuck closed, start cap will be blown, often exploded.
If you have an amprobe, let me know and I will give you THAT method to check out!
Confused yet? That's OK as more than a few techs have trouble with this part of the job!