Best way to travel by bus or train from London trafagler square area to Dover to walk the cliff.

Tube to St Pancras then train from St Pancras to Dover (journey around 70 minutes).

A new Monday to Saturday bus service #66 starts operation on 8 December from Dover Railway Station to the Eastern Docks (Dover Ferry Port).
It runs hourly from 0720 to 1120 and 1420 to 1720, returning from the port at 9 minutes past the hour from 0709 to 1109 and 1409 to 1709.

The most economical way to do that would be to buy a Dover Plus Bus ticket with your rail ticket for £4.50 each or £2.95 each if you have a national railcard- https://www.plusbus.info/dover

There are National Express buses from London Victoria to Dover Pencester Road (where the 66 also stops) but they are a lot slower, and unlikely to be cheaper than Advance (train specific) train tickets.

Thanks for the info.