How far do I have to walk from London underground station to St Pancras international station?

Kings Cross St. Pancras underground station has exits directly into the St. Pancras International station complex.

The Eurostar departures area is in the part of the station that is closest to the Metropolitan/Circle/H&C line platforms.

Expect to walk 150-100 meters after you exit the Underground ticket hall. That's after up to 100-200 meters in corridors to get from the platform to the station exit, depending on which line you arrive on. There's a diagram of the passages at https://tomhalltravel.files.wordpress.com/2009/12/king-cross-upgrade-map2.gif -- if you're arriving on one of the deep tube lines, first follow the signs towards the Circle Line until you're in a ticket hall, then exit through the ticket gates and follow signs towards St Pancras.

(For actually catching your train, remember that there's a check-in procedure and you may have to wait in line to pass the French immigration controls before you get to the platform. So walking is not going to be the main item in your time budget).

The Kings Cross St Pancras underground station is linked directly to the St Pancras International train station, you won't see the light of day while going from one to the other.

If you arrive using the Circle, Metropolitan or Hammermith & City line, the station is just under the front of the station, and it's about 200 meters from underground platform to Eurostar check-in.

If you arrive using the Northern, Picadilly or Victoria lines, then those lines are pretty deep underground and it takes slightly longer (count 250-300 meters).

Also, the signage on some of those platforms now wants you to take the new route they added a few years ago, which is accessible, but much longer, as it brings you all the way to the "back" of the station, and it's probably closer to 500 meters to Eurostar check-in. It may make sense if you're not travelling by Eurostar but using one of the domestic services, though.