Take the Heathrow Express train from London Paddington to Heathrow Airport. It is cheaper and quicker than a taxi. The ticket costs �14.50 bought online, �15.50 at the station, or �17.50 on the train.
To get from London Euston to London Paddington, either walk for eight minutes and then go on the underground for nine minutes (�4 cash, or �1.50 with an "Oyster card", or take a taxi.
A pre-booked taxi from London Euston to London Heathrow will cost �35.
From the Heathrow Airport website:
"The approximate cost from Heathrow to central London is �46-50 with an average journey time of one hour."
Another way of looking at it:
A taxi from Liverpool Street to Waterloo, which is about 2.5 miles miles, usually costs me about �12. So the journey from Euston to Heathrow (which is about 18 miles) will cost substantially more.
The cheapest way to do the journey is take the Victoria Line to Green Park and then transfer to the Piccadilly Line for the journey to Heathrow. (�4).