Can I upgrade to i7-7700k, the newest Intel CPU?

First here is the IFIXIT guide to get to the CPU: iMac Intel 27" Retina 5K Display Logic Board Replacement Jump down to Step 61 to see how far you'll need to go.

I should point out if you haven't removed the display on the newer 'Thin Series' iMac's it's a bear! And if you don't use the correct tools as well as follow the IFIXIT guide to the letter you can easily damage the display! Which would be an expensive opps!

As to what CPU your system can support the best that Apple used in your series was a 4.0 GHz i7-6700K it uses a FCLGA1151 socket and has a TDP of 65W

The new Intel kabylake 4.2 GHz i7-7700K CPU will likely not to work in your system. It does still use a FCLGA1151 socket but its TDP is higher 91W. Which your system was not designed to support.

I don't know of anyone trying it yet as its just coming to market. Frankly, I don't think its worth the risks, it would be safer to just get one of the newer iMac's and sell yours to recoup some of the costs if you really need the slight added performance. Here's a quick review: Mini-review: How much faster have high-end iMacs gotten in the last 5 years? That might put your mind at ease.

If you need the horsepower you'll be better off waiting for the iMac Pro or the still coming newer Mac Pro!