Q: Power Mac G3 no output to screen

Hmmmm, this is going to be a bit tough remotely with not much to go on, but...

Open it & see if the graphic card has come loose, maybe reseat it.

Do you know which G3 it is... there were ones called Beiges, then ones called Blue & Whites.

I understand why there's not much info available, no problem.

When it starts up do you hear a bong sound?

Anytime that you buy an old computer like that, you should assume that it was probably disconnected from electricity for some time.  Depending on the seller, it may have been in storage for months, or been on a pallet-load of old computers bought at auction.  When disconnected from electricity for long periods of time, the battery on the motherboard becomes discharged, especially if it's an old one.  It's a 3.6-volt, ½AA battery, as seen here.  A very weak or dead battery can cause a number of startup problems, including no video to the display.  Was the date & time set correctly, after booting it up for the first time.  If not, it should be replaced.

The use of a third-party keyboard, mouse, display, speakers, etc. will not cause the symptom that you're experiencing.

I'd get a fresh PRAM Battery, which was my next suggestion but Jeff beat me to it.

Those batteries are not common. You may have to pay a premium at a special battery store or Radio Shack. If you have time the last time I needed one I ordered it from mac-sales.com/OWC for a couple of $

I was going to second Limnos suggestion. OWC has the batteries for around $4 plus shipping. Radio Shack has them for $19.95. I'd go with OWC if you haven't already. Best of luck on the Mac.

And since the new PRAM  Battery did you do 3 PRAM Resets in a row?

PRAM reset, CMD+Option+p+r...

http://support.apple.com/kb/ht1379

In fact, do 3 in a row, takes a bit of time.