I had a look around, and if you go here:
https://www.notebookcheck.net/Intel-HD-Graphics-630.187948.0.html
Scroll down about half way and it'll give you some game examples.
The 8700k will be maybe as much as 20-25% better since these are laptop CPUs tested, so have lower clockspeed. But it looks like it'll only be good enough for low settings at 1920x1080 (or lower res) on anything other then very easy to run games.
I couldn't really find a comparison to the 4870. The 4870 is likely faster still, as it should be around an RX 450. The 4870 is probably a bad idea to use anyway though as it only supports up to DX10.1 and OpenGL 3.3. So regardless of if it has the processing power, you may not be able to play some modern games on it anyway.
What kind of work do you do other than gaming? You may not need the CPU power of the fastest 6-core money can buy. It may be better for you to wait for the AM4 Zen+Vega graphics APUs, if you want competent CPU performance but also gaming in one package. Although we don't have an exact date for that, and you'd only really want that if you also wanted to save money.
If money isn't a big deal, the best bet could be go with the 8700k like you plan, but then get something like this:
https://www.overclockers.co.uk/giga...ddr5-pci-express-graphics-card-gx-19e-gi.html
That card will only spin up the fans when it goes over 60 degrees or so. So will remain totally passive unless you play a demanding game. It is also very low power, so doesn't require any power connectors, just slot it into a PCI-E slot. And also has the processing power to run any modern game at 1920x1080 and medium/high.