To diagnose a non booting asteroids you need to do the following:
1. Verify that all roms are good. Do this with a eprom programmer and the relevant mame rom set. If you are using the high score kit, then that would eliminate the three program eproms (assuming you have revision 2) but it doesn't eliminate the vector roms.
Like the previous posters have mentioned, you should ideally check/replace the CPU socket and Vector Rom socket if you suspect anything. They are known to go bad all the time. A reasonable method of testing if your sockets are good is to hold the probe on reset pin 40 and simultaneously lightly press down on the chips. If you notice the reset pin changing then it's likely a bad socket.
2. Check pin 37 for clock signals. If you are getting rapid pulsing then it's likely your clock signal is good. If not you should check the clock circuit. This consist of mainly C4, B4 and B5.
3. Check pin 40 for reset. This should tell you a lot to what is causing the problem. Thing to note is that the reset pin should momentarily go low on power up and then stay high if working. This is necessary to get the CPU to boot. If you find that this doesn't go low you should check the reset circuit for a fault. The following chips should be checked for correct logic function. B6, C6, D5, E6, D4.
4. If the reset pin is pulsing high and low rapidly, there is a high chance there is a problem with your address circuitry. Before that check the buffers at B2 and C2. Check data sheet on those two buffers and check logic signals to see if they are working as normal.
5. Address circuitry consist of the following chips. E4, D6, E5, C6, E4 and L3. You will need to probe these chips. If any of these chips are behaving badly it will cause the game to not boot. Again use schematics and datasheets to check if these chips are behaving as they should. Look for floating pins, stuck pins, pins that should be flip flopping when they are not. Socket and replace any chips you suspect to be bad.
6. Once all this is done, the game should theoretically boot to at least test mode (you still may not get sound/video) where it will tell you if the ram is working.
You should do all this without the CRT plugged in to be honest. You are going to get a horrible hole in your CRT if you don't. There is no need for visuals at this stage.
Once you get to test mode you can then hear what else needs to be fix.
HTH