One way to check the computer itself is to plug your new monitor into it and try booting up. If it starts to boot up but quits, make note of where it quits and any error message that might be showing on the screen...that will give you some info to post here so we can help determine what needs to be done in order to get it working right again.
If it doesn't boot up at all (no lights, beeps, sounds of fans, etc.), it could be that your power supply at the top back of the computer has finally given up the fight and needs to be replaced in order to get the computer working again. They are easy to replace and pretty inexpensive, but you would have to check inside the case to see if it's an AT or ATX type....we can tell you how to know the difference.
I now have an Window's XP. But my old computer used to be a Windows 95. And one day we started it checked email and then played freecell. We left for about 2 and half hours or 3 and when we came back, and the screen went blank we figured that it was the sleep mode but the button was broken so we had to shut it off manually we tried not doing that very often but it used to freeze all the time so we had to. So we figured well just do it that way but then when we tried to turn it on it just acted like it was then it was shut off. We had it about 7 years it was a Compaq and i was wondering was it because of the age, It crashed, or some virus got us. or If The Company Compaq has faulty computers or something and if so please help me because we got another Compaq because we figured that it lasted that long. that we buy another one just the same company. Have Had it about 10 Months.
SO if anyone knows anything please help ME.!! Oh it was just a plain and simple old Compaq Windows 95 computer that we bought in 1996. Thank you.