Travel from Dublin, Ireland to London, UK

Ronald, The "best & fastest" way will likely be a budget air flight using RyanAir or possibly Aer Lingus. You'll have to check their respective websites for current prices and other details. Be sure that you read their Terms & Conditions VERY carefully, especially related to luggage weights. RyanAir is infamous for the "extra fees" they charge, so the price you actually pay will likely be somewhat higher than it first appears. I try to avoid using RyanAir unless I'm dreadfully desperate, so if I were in the same situation I'd take the short (~2 hour) train ride to Belfast and use an EasyJet flight to London Gatwick / Luton / Southend / Stansted. They currently have flights listed as low as £25.99. They also have a few extra fees, but they're more reasonable (IMO). Happy travels!

Ryanair has a poor reputation, much of it generated for marketing purposes by its CEO - any publicity is good publicity, right?), but if you don't want to fly with them or Aer Lingus then the third option for Dublin to London Heathrow is British Airways. I just found a few flights on 19 June from €69 on their website, but they are selling fast it seems.

There is also Air France, flying Dublin to London City. On any given day at you desired time one will be the cheapest but it could be any of them, all charges taken into account. I happen to be in Dublin now and flew from Heathrow on Aer Lingus but will return on British Airways.