There are a number of ways to travel to Dublin from London. The easiest and probably quickest way is to travel by air.
You will need to check which of the three airports in London fly direct to Dublin. The airports in the capital city are Heathrow, Gatwick and Stansted. The Irish airline Aer Lingus flies from both Heathrow and Gatwick so it's possible that you could get a good deal on a flight through that airline. Depending on the time of year and the time of booking it is possible to get some really good flight deals for a return trip to Dublin.
Other travel alternatives could be by the train, ferry and car option.
Trains leave London Euston and passengers can either travel to Holyhead in Wales or Stranraer in South-West Scotland. Once there a ferry can be caught to either Belfast or Dun Laoghaire in Ireland. A train can then be caught to Dublin from both of these destinations. If passengers take the Holyhead to Dun Laoghaire route, on arrival at the ferry port a taxi or a bus to Dublin will be an inexpensive form or finishing the journey as Dublin is only eight miles away. The train option for this journey could possibly be the best option if you are on a budget as you would pay a foot passenger rate on the ferry. Also, there are some good deals to be had by booking train tickets in advance of your journey. It's always best to forward plan and try to find the cheapest options for any travel.
The same journey can be undertaken in the car, but it would be easier for the driver to head to Holyhead instead of the longer haul up to Stranraer. Extra charges are incurred on the ferry for crossing with a car, but once at the other side it's a straight forward eight mile drive to Dublin city.