You have a flaky PSU that's not getting the output out good enough or the "uuppphmmm" to get the PC up and running. If you get a replacement PSU, besides one that fits, add more wattage to it, if you have 350W PSU, having a 450W or better is more than enough. @ yrs. it maybe showing the wear&tear of operation and starting to falter and repeated reboots gets it going.
Also, you may have USB devices besides kybd. or mouse that drags down the PC from starting correctly. The message you're getting, "no signal" is really OK for the monitor as it awaits for the PC to provide it, which if not booting-up is the result. You may want to wait abit longer and see if the PC actually boots-up on its own after all.
It wouldn't hurt to clean the PC insides as well upon opening the PC case. Remove the dust bunnies, as "heat stress" can effect the PSU over time and result in a weaker PSU, which in turn can't provide entirely what the PC needs to start cleanly.
My desktop is an HP p6604y with an AMD Athlon II 220 processor, 6 gb ram and is a 64 bit system.
This machine is about 2 years old, and many times when we start it up, it will take 3 power button hold-down restarts to get the screen to come on- screen will indicate no signal and go into sleep mode. We have had the machine in 3 times to local computer shops at about $50 each to which they can either find nothing wrong, or think they fixed it. Further repair shops seem like a waste of money.
We have: turned off all screen savers, power/energy savers, checked cords, switched monitors, etc. and nothing holds.
This is my last ditch effort to figure something out before I either learn to live with it, or get a new one. Any help or suggestions will be appreciated.