Computer 1 : Computer screen goes black? can you get into Bios?
Does it keep Restarting every 5-7 seconds??
If the computer seems to stay on, but your just getting a black screen, then...
I would double check to make sure your cables are inserted all the way ( Expecially if its HDMI ) .
If HDMI, sometimes you might have to wiggle the Graphics card up or down just a tiny pinch, so that the HDMI cable can actually plug in all the way,
and not be blocked by the Faceplate ect ect.
But if the computer turns off instantly or continues to reboot every 5-7 seconds, you possibly might need a new CR2032 Battery, for your motherboard.
Normally you need to replace these every 4-7 years.
Computer 2: Try logging in, with and without internet connected, see if it makes a difference.
Can even try running SafeMode with and without networking.
Depending on how old the computer is, it might be helpful if you run a DISK cleanup, and Defragment.
Might be better off just installing a fresh copy of windows 7, or maybe even 10, if the computer is fast enough.
( Keep in mind ) , That even the fastest computers are slow for the first 30 seconds after logging in.
You can open Task manager > Startup , and disable some of the un-used programs from starting up right away, set delays, or disable startup ect.
Old computers that uses windows XP can take 5-7 minutes after logging in, before things stop being slow at start.
Hope this helps, maybe provide us with more details / pictures ect ect.