Maybe worth trying an Ubuntu live CD without a HDD/SSD in place and try the CD/DVD drive through different SATA ports. That would eliminate another possible source of the problem.
But you are probably at that point where you either have to replace the power supply or the CPU or motherboard. The cheapest option would be to buy a really cheap power supply to see if the PC boots with a minimum amount of hardware - £15 would get one. If it proves the power supply is faulty you have only spent a few £ and could then get a more suitable one. If it doesn't boot you then have the issue of it potentially being a motherboard or CPU issue. Given the temp issue, it could be that the CPU went faulty causing the high temps. I'd probably be tempted to get a new CPU/motherboard combo rather than risk the existing pair but obviously depends on the budget after testing power supply.
Found this which may help. May be a little old as when was the last time you used a floppy disk?
Symptom Check
When I turn it on...
It makes no sound and there are no lights
•Check the Power Supply
•Check the Motherboard
•Check the CPU
When I turn it on...
It just beeps
•Check the Beep Codes List
•Check the Keyboard
•Check the Video Card
•Check the Memory
•Check the Motherboard
When I turn it on...
It sounds normal but the screen is blank
•Make sure the monitor has power
•Check the connection to the video card
• Check the Video Card
When I turn it on...
It starts up but never loads the operating system
•Check the HDD/SSD
•Check the Memory
•Check the Motherboard
•Check the CPU
When I turn it on...
It freezes BEFORE the operating system starts loading
•Check the Keyboard
•Check the Video Card
•Check the Memory
•Check the Power Supply
•Check the Motherboard
•Check the CPU
When I turn it on...
It freezes WHILE the operating system is loading
This is probably a software problem. If you are using Windows, try booting into safe mode by holding down the "F8" key or in some cases the "CONTROL" key.
While it is on...
It makes a screeching or other loud noise
•Check the Power Supply, CPU, Case or other Internal Fans
•Check the CD-ROM
•Check the Hard Disk
•Check the Floppy Drive
While it is on...
It shuts itself off
•Check the power features in the BIOS
•Check the power features in the Windows Control Panel
•Check the Power Supply
•Check that all fans are working
•Check for the accumulation of dust