To get to Stamford Bridge you catch the Piccadilly
line from Kings Cross to Earls Court. You can then either hang about for a train
to Fulham Broadway (could be quite a wait), or walk (follow the signs for
'Exhibition', then turn left out of the station). You'll need a Travelcard for
zones 1 & 2. It's about a 15 minute walk to the ground. You'll be lucky
to do it quicker by waiting for the tube!
For years we used to go to the Fox and Pheasant for a pint. It's a cracking pub - the look and feel of a country pub, but in London. And it's only a five minute walk from the ground! However, they now don't allow away fans in. When we played down there in the relegation season some of our lot ended up in the Fulham Tup. At the time it got pretty horrendous reviews. However, it's since closed and re-opened as the Rose, and sounds a lot better. Your best bet could though be the White Horse. This is far enough from the ground not to get too busy, and allows away fans in. To get there you'd stay on the tube to Parsons Green, then the pub's just around the corner. Another option is the Goose (nearest station West Brompton), which has the added benefit of being very cheap! Nearer the ground the Slug and Lettuce, right outside Fulham Broadway station, apparently usually gets plenty of away fans in. The George, a short distance along Fulham Road, is also ok for away fans. There's a Lloyds No 1 (the Oyster Rooms) in the shopping centre above Fulham Broadway. It gets decent reviews, but I've no idea if away fans can get in. To be avoided are the Sofa, the Britannia, and One. All the pubs mentioned are marked on the pub map below.
Kings Cross pub map
Internet Ground Guide