Usually you replace the parts that smoked. If it is years old, I'd move to a CPU and motherboard combo and hope the rest comes along.
To test the PSU you need a Volt meter (about 10 bucks.) I take it you have the basic repair gear?
I have a desktop I built myself a couple years ago, an AMD build. Today I was in the middle of gaming when my computer suddenly shot off. I tried turning it back on and did for maybe a second then shut back down and smoke was coming out of it . It appeared to be coming from the CPU heatsink. I opened the computer up and checked the mobo over, it all looked fine. I've tried turning it on again after letting it cool off but the only thing that turns on are the lights on the front of the case. It doesn't boot up at all, no fans or anything.
I assume I'll need to replace something but I'm unsure if my issue is the psu or maybe the CPU heatsink and/or CPU itself?