Sounds like bad caps on the PSU motherboard. I'd bet my bottom dollar.
Agreed. I mis-read this and started typing up a different response (along the lines of "black screen + known battery leakage = probable corrosion in the CPU socket"), but I'd worry about the PSU first.
Make sure you have a clean and functioning power supply before moving on to the motherboard issues. Easy option would be to just swap it for a replacement, but obviously we don't know your budget or skill level, or if you have available spare parts to swap with, etc. OP needs to be prepared to post some pictures and do a lot more troubleshooting if they hope to solve this themselves.
