Nothing different than what has already been suggested really--try it in another system that is currently working. I know my 3770k works and the exact setup under which it works and I've dealt with marginal cpus before that have had some sort of issue, so I have a hunch it's not bad or I can save it.Can't you just share any ideas you got, that we may not know to help fix it?
One thing I would look at that I have seen before is check the corners of the cpu to see if they are bent. I've seen this on an i3-2120 that was acting funny (very similar scenario to yours), and one of the corners was bent upwards. I don't think the chip is completely bad, but it would do a boot loop after getting past the bios. And later would power off 2 seconds after power on--the fans would barely get up to speed. I ended up replacing it with an i5-2500k.