Would you happen to have a recommendation for the thermal compound remover I can use for removing the old CPU?
Well, you remove the screws from the heatsink, which is copper & kind of long, and then gently wiggle it from side to side a bit to break the seal, you'll feel it, then lift the heatsink out (make sure that any fan connectors are disconnected), and I suggest that you take pictures along the way. A digital camera is best, though some cell phones works good as well.
As to the cleaner, 91% rubbing alcohol at most any store for under $2, and if you have any, a couple of coffee filters for the final cleaning. Though you may use any lint free paper towels first to get the bulk of the old paste off. Q-Tips are also good to have to work in crevices.
First off, go ahead & clean the heatsink up good, remove all traces of the old thermal compound. There's likely dust in the fins, so be sure to blow these out with air dusters designed for this purpose, preferably outdoors.
Once you have it good & clean, now take one of the coffee filters & soak with the 91% alcohol & rub the CPU & GPU contact (if applies) well, you'll see that the paper towels didn't get it all. Do it until no more gray shows & allow to dry. Now for the CPU. Before you remove it, clean it up, though don't get lint everywhere, do it more gentle than the heatsink & note the position of CPU before removal. Raise the lever that secures the CPU (you may have to pull the locking lever over a bit for it to raise), and using a Q-Tip, clean the excess thermal paste from the CPU and MB. There should be a notched screw (note position), turn it a quarter to half a turn gently to raise the CPU for removal & set it in a position where no pins will be bent.
Make sure all is clean in the area, sometimes thermal paste may go beneath the CPU at bit (a sign of too much applied). Now place the new CPU in the socket & turn the screw back exactly the way it was to lock it down. We're now getting ready for the install. Take one of the coffee filters & clean the surface of the CPU good, though more gentle than with the heatsink, keep cleaning until no more residue shows. Allow to dry 10 minutes, if you have a discrete GPU (you'll know because there'll be a place on the heatsink for it that you cleaned), place a small drop on it (very tiny), then an oat or rice grain size drop on the CPU. Now lower the heatsink into place, and wiggle a bit to make the paste spread. Take the screws, and in a criss-cross pattern, tighten the screws in three rounds, the first should be just sung, the 2nd & 3rd is just making sure it's going to say in place. Don't over-tighten, you'll take a chance of cracking the GPU or damaging the MB.
You want the heatsink to stay in place, but don't need to torque it like a set of lug nuts on rims. Just snug it down good. Now reassemble the rest of the notebook & hopefully it'll boot.
If not, you may have a motherboard issue, or a backlight one (the screen isn't working due to the inverter needing replacement).
When I power the laptop on in its current state, it sounds like the hard drive, fan, etc turn on but it's all so quiet I cannot tell for sure.
More evidence pointing towards an inverter.
I found a refurbished motherboard for $47 on amazon!
You mean you likely found a used motherboard on Amazon, someone else's troubles. This is one item that I'll never purchase used, for starters, to fully refurbish a MB would cost a lot more than $47 in parts alone. There's the DC jack to replace, make sure there's no bulging capacitors, no loose or corroded PCI/PCIe/RAM ports, and all connections like new. I seriously doubt at $47, the MB has been refurbished, which is why I stated likely used instead.
Too many sellers on Amazon/eBay declares their components as refurbished, when in fact all these received, if that, was a cleaning.
Hopefully the CPU will do it, though seldom do I see one causing video errors.
EDIT: Correct spelling typo above to say 'dry'.
Do it until no more gray shows & allow to dry.
Edited by cat1092, 05 October 2016 - 01:47 PM.