Replacing thermal paste and cooler in Intel i5-2500K

Yup just leave the CPU in the socket and clean it there, I do that all the time and is the preferred method.

You can use coffee filters or paper towels, but coffee filters leave less fibers. I believe any type of rubbing/cleaning alcohol is good enough.

What is your idle temperature?
I expect to see 10-15c. over ambient with the stock intel cooler.

70c is a very normal temperature to see under load.
The cpu will throttle or shut off if it detects a dangerous temperature.
That is around 100c.

What is your case?
What is your fan arrangement?
Any cooler needs a good source of fresh air to do it's job.

When replacing a cooler leave the cpu in the socket.
A socket has a design point of 15 insertions, so do not remove the cpu without cause.
You can easily damage pins if you re insert badly. Not good.
If you heat up the cpu by running it before the job, the old cooler will come off more easily.
If you touch the case before working, you will discharge any residual static.
I use paper coffee filters to do the cleaning job.
Mainly because they are lint free.
I tear off a piece and put a bit of alcohol on the piece and wipe of some of the paste from both the cpu and the cooler.
Repeat with new strips until you get all of the paste off.
It will take perhaps 5 passes until the last one is completely clean.

All paste is ok.
AS5 is one of the ones I use, but the paste that comes with a cooler is ok too.

Be careful about using too much paste as it can act as an insulator.
A rice sized drop in the center will spread under heat and pressure.
It is hard to use too little.
If you are going to overclock the 2500K(and I think you should) then a better cooler is in order.
Assuming you have 160mm available for a cooler like the hyper212, then I recommend the scythe kotetsu@$35.
https://www.newegg.com/Product/Product.aspx?Item=13C-0004-00005
I was impressed when I installed one in my son's pc with a 7600K overclocked.
Very quiet.
Here is a review:
http://www.silentpcreview.com/article1391-page1.html

In the event you want to try reinstalling the pushpin cooler, here is a copy of my stock instructions:
----------------how to mount the stock Intel cooler--------------

The stock Intel cooler can be tricky to install.
A poor installation will result in higher cpu temperatures.
If properly mounted, you should expect temperatures at idle to be 10-15c. over ambient.

To mount the Intel stock cooler properly, place the motherboard on top of the foam or cardboard backing that was packed with the motherboard.
The stock cooler will come with paste pre applied, it looks like three grey strips.
The 4 push pins should come in the proper position for installation, that is with the pins rotated in the opposite direction of the arrow,(clockwise)
and pulled up as far as they can go.
Take the time to play with the pushpin mechanism until you know how they work.

Orient the 4 pins so that they are exactly over the motherboard holes.
If one is out of place, you will damage the pins which are delicate.
Push down on a DIAGONAL pair of pins at the same time. Then the other pair.

When you push down on the top black pins, it expands the white plastic pins to fix the cooler in place.

If you do them one at a time, you will not get the cooler on straight.
Lastly, look at the back of the motherboard to verify that all 4 pins are equally through the motherboard, and that the cooler is on firmly.
This last step must be done, which is why the motherboard should be out of the case to do the job. Or you need a case with a opening that lets you see the pins.
It is possible to mount the cooler with the motherboard mounted in the case, but you can then never be certain that the push pins are inserted properly
unless you can verify that the pins are through the motherboard and locked.

If you should need to remove the cooler, turn the pins counter clockwise to unlock them.
You will need to clean off the old paste and reapply new if you ever take the cooler off.
----------------------------------------------------------------------------------

I've just (as in 2 hours ago) upgraded the stock cooler on my 2500k running on an Asus P8-z68v Pro/Gen3 with a Corsair H100i V2, I was idling at 45C on the stock cooler and any stress test i ran had it hitting the 90C mark and thermal throttling badly.
With the Corsair unit installed it now idles at around 28C and an hour long stress test had it maxing out at around 49C.
Room temp was 24C.
I removed the CPU and used Isopropyl alcohol w/ cotton buds to clean the old thermal goop off, and them just used the preinstalled Corsair thermal pad.

Just thought i'd give you an idea of what my experience was.

Yeah, i have a Corsair 650D and it fit really nicely into the top mounting brackets.
Did you sort your cooling out?
How does the 1070ti work in your config? i'm looking to replace my 6950 (still works fine but the new gen games are giving it a little bit too much trouble.

Gotta say though, they do run and that little 2500k is still a champ 6/7 years on.