1 = The CPU being still hot minutes after you turn the PC off, may indicate the cooler fan is not working or disconnected, thermal paste is insufficient, or not applied. Make sure none of these is the case. A hot CPU may shut the computer down and if your fans are silent, you may not have noticed it happening, so make sure if that happened or not. Also check if all fans and led lights are turning on.
2 = Next thing to check would be the BIOS. Reset it as follows:
a) Remove the AC cable.
b) Remove the battery
c) Move the BIOS jumper from pins 1 and 2 to pins 2 and 3
d) Wait 10 minutes before reverting everything and starting the computer
3 = Would be a BIOS Recovery.
a) Remove the AC cable.
b) Remove the battery
c) Remove the BIOS jumper
d) Press the Power button and hold it pressed for a minute
e) Reconnect AC cable, install BIOS battery and jumper and start the computer
Hopefully 1 to 2 may solve it.. but in case that doesn't happen, check these possibilities:
4 = 1 and 2 failing, did you have any trouble installing the new CPU?.. if so, the socket pins are sensitive and it's easy to bend them if you're not extra careful.
5 = I've seen and read that PSUs sometimes fail when you do unrelated modifications, upgrades, and even when simply moving the PC. Try a diffent PSU or test this one in a different PC.
6 = Was the i5 CPU having any performace issues? If that was the case, the issue may be either: the PSU, lose RAM module, motherboard issue, lose PSU connector on motherboard.