We used to regularly rebuild the power supplies and replace the logic board capacitors. Here's a video on removing the logic board capacitors in three minutes: http://www.youtube.com/watch?v=5VjtzugrIL0
We used to be able to buy the capacitor repair kits for $15
Here's a very good article on diagnostics and what to repair on an SE/30 which will work fine for the classic. http://www.jagshouse.com/classicrepair.html
Just watch out for the end of the CRT as it will break easily and also knock you on your butt if not discharged.
Here's a gutsy logic board cleaning method I have not tried: http://www.youtube.com/watch?v=idnD9DHFUic&feature=related