I make this journey quite regularly. Here is my recommended route
cross Euston Road
Upper Woburn Place;
Tavistock Square;
left into Tavistock Place;
follow the two-way cycle path (beware traffic crossing from the right and be aware that there are separate lights for cyclists at some of the junctions);
continue on the cycle lane/cycle route through Regent's Square and Sidmouth Street until you get to Kingsway College;
right onto Gray's Inn Road;
continue past
Condor Bikes and straight on at next set of lights;
left onto High Holborn;
continue on to Holborn Viaduct (watch out for people turning left without signalling at the roundabout by Hatton Gardens and the Sainsbury's Head Office);
continue on Newgate Street passing the Old Bailey;
as you get to the lights by St Paul's tube (just opposite the bombed-out church) filter right then turn left onto the (one-way) King Edward Street, then first right (hence need to filter right) onto Angel Street (marked buses only) then right again onto St Martin le Grand;
you need to cross St Martin le Grand (not as scary as it first seems - drivers are normally very good at giving way);
left onto Cheapside;
continue along Poultry until you get to the lights by 1 Poultry (big James Stirling building with a branch of Tesco on the ground floor);
right onto King William Street;
at the traffic lights at the junction of King William Street position yourself on the right;
turn left then sharp right onto London Bridge;
at the end of London Bridge take the second turning on the left (the first turning on the left, by
Evans cycles and the spiky thing is Tooley Street) and into the station.
Trains for Dover usually leave from Platform 5.