to the ground
To get to Stamford Bridge you catch the Piccadilly line from Kings Cross to Earls Court, from where it's the District line to Fulham Broadway. It should take about half an hour. It's then a 5 or 6 minute walk to the ground. Alternatively, you could just walk from Earls Court (20 minutes), particularly if you opt to drink in one of the pubs around there.
All of the pubs within a mile or so of the ground are seem to be strictly home fans only. Most away fans tend to stick to the pubs around Earls Court tube, before making their way to the ground late on. There are quite a few to choose from. but they're all pretty upmarket, with fancy/expensive food. Three of the nearest are the Courtfield, the Blackbird, and Prince of Teck, while the Kings Head is tucked away down a side street so should be quieter. Alternatively you could head for the White Horse at Parsons Green, which gets really good reviews. You stay on the tube for one stop after Fulham Broadway, and obviously have to come back again later on. A final option is to head for Victoria (only 8 minutes from Kings Cross), where there's a Spoons inside the station, and then head for the ground at the last minute (15 minutes to Fulham Broadway, but 10 minutes between tubes).
Kings Cross pub map
Chelsea FC visiting supporter guide
Football Ground Guide