Thermal paste has nothing to do with whether or not a laptop powers on initially. Way too much paste or no paste can cause overheating, but it should power on at least initially and stay on for a minute no matter how well or badly you applied the paste. Once the board is really hot, it will shut off in less time, i.e. the 5th time you power it on it might shut off in ten seconds, since it's already warmed up.
Therefore, the board installation is most likely the culprit. Take the board out and re-install it carefully. Make sure you are using the correct screws in the correct places and nothing is getting shorted.
As far as thermal paste, you only need to use a tiny drop the size of the head of a small screw. Any more than that, and the paste will spread too far and conduct heat where it shouldn't go. Make sure the previous paste has been completely removed and cleaned away with alcohol.