Agree that the easiest way is train via Euston. Now, getting to Euston. If you are traveling light (say one roll-on) and you can carry the baggage up a small flight of stairs, the quickest ways would be -
1) Quickest, but expensive:
Take the Heathrow Express to Paddington. Trains run every 15 mins from terminals 1/3/5 and take 15 mins. Then take the Hammersmith & City line tube to Euston Square. Note the H&C line operates from the high-platform numbers of Paddington's main rail station concourse. It's about 6 minutes to Euston Sq. Total cost, 22 pounds (18 for the Heathrow Express)
2) Longer, but cheaper (and definitely recommended if arriving at T4):
Take the Underground to Hammersmith Station. Journey time about 30 mins, but trains run every 5 minutes. Change to the Hammersmith & City or Circle Line, journey time about 20 minutes. You will have to leave the station at Hammersmith and cross the road to get to the H&C/Circle platforms, but it's only a short walk. Cost, around 7.50 (safest to get a zone 1-6 travel card because of the change at Hammersmith).
You'd have to climb the stairs at Euston Square (no escalator). This station is 50-100 yards from Euston mainline station.