When diagnosing an old PC, I always take the step-by-step approach. First, take out all the expansion cards, and set the DIP switches for the minimum configuration: no floppy drives, no video, and just the base RAM (either 64K or 256K, depending on the revision of your motherboard). Power it on, and after the memory check, you should just hear a single short beep from the speaker, indicating that it has booted up directly into Cassette BASIC.
Then, add in the display adapter, set the video DIP switches accordingly, plug in the monitor, and power it on. Now you should be able to see what's going on. If not, there's a problem with either the display adapter or the monitor. (Also, make sure you are using the 5151 with an MDA card -- it will plug into a CGA card, but won't display anything, and the wrong video signal type may even damage the monitor!)
If you don't have another monochrome monitor or MDA card to try, you can swap in a VGA card and use a standard VGA monitor -- as long as your PC is the later BIOS version, usually found on the 64/256K version of the motherboard (which is the most common). Most 16-bit ISA video cards will work fine in an 8-bit slot. Set the PC's DIP switches to "No video", which is the setting used for VGA or EGA.