Please help me find replacements for my Mobo and CPU!

So here's the story: one day, my pc refuses to boot. No bootable device found. Okay, first thought is obviously this is the SSD. RMA'd, no service as no fault found.

So then I remember I've had BSOD's related to my GPU. I figure this must mean something else is at play. Now I believe it's the motherboard at fault but I don't think I have a way to diagnose if it is the mobo and CPU. My backup HDD spools up so I assume its not the PSU in play.

It seems most efficient to just upgrade both as opposed to one or the other as both components are fairly old.

Current specs: i7 3770k (Ivy Bridge)

Zotac GTX 780

Gigabyte GA-Z77-D3H

Samsung 850 EVO 250GB

OCZ700MXSP

Corsair Vengeance 16GB - I don't remember the clock speed!

The PC was originally constructed in 2011 and had a GPU replaced since then. (680->780)

Budget: As cheap as possible but for equivalent performance to my 3770k at least. I don't need fancy features on the mobo, it needs to work. I am based in the UK.

Use: Studying, video games (mostly slightly older games but I also play Warframe, would like to play Overwatch, League of Legends, etc)

Thanks very much everyone!