The Football Association of Wales confirmed Coleman's appointment until 2014 at a Cardiff news conference on Thursday.
Coleman said it was one of the proudest moments of his career but also a difficult situation.
"We're still all shocked by what happened [with Gary Speed] and we're still grieving," said Coleman.
"And the only way we can put smiles on fans' faces again is to continue to win matches but I don't think we'll ever get over the loss of Gary.
"I've already spoken with the FAW members... and I said 'excuse me if I don't seem that excited, I am excited, I'm just a little bit subdued because of the circumstances'.
"It won't be easy but I'm looking forward to it. I think for any manager who was going to come in it was going to be difficult because of the situation and it's more difficult for me with my relationship with Gary.
"...It's not going to be easy, but it's something I'm prepared for.
"On one hand it is the proudest moment of my career but on the other hand, with the circumstances I was very close friends with Gary for 30 years.
"It is the most difficult press conference I am ever likely to do."
The FAW's chief executive Jonathan Ford admitted it had been a difficult time since Speed's death in November 2011.
"We have explored every avenue, every scenario possible," said Ford.
"We have taken on board the thoughts and views of the backroom staff, the players and the supporters.
"All along we believe that we have been respectful and dignified and had no wish to be involved in a public debate on the appointment of Gary's successor.
"Gary was the manager of Welsh football and of course we want to continue his legacy.
"He was not a figurehead, he was the manager and that is why we have decided on the appointment of Chris Coleman to build on the excellent foundations put in place by Gary."
Former Wales international and Manchester City manager Mark Hughes added: "I think he'll [Chris] do a great job and respect the legacy Gary left behind."
Former Fulham boss Coleman was contacted by the FAW last week after resigning as manager of Larissa and citing financial problems at the Greek club as the reason for his departure.
However, on the day Coleman was appointed Wales manager, Larissa began legal action to claim compensation from the Football Association of Wales.
John Hartson, another former Wales international, was the only other candidate to publicly announce his interest, while Ryan Giggs and Ian Rush had been suggested as possible "figureheads".
Coleman joined Larissa in May 2011 as he attempted to resurrect his managerial career after a year out of the game.
His last job in England was with Coventry City, but he was sacked in 2010 after a difficult two and a half years in charge.
Before joining the Sky Blues, a promising start as manager of Real Sociedad ended in resignation following internal upheaval at the Spanish club.
Injuries suffered in a car accident cut short Coleman's playing career at the age of 32. His last match was a brief appearance as a substitute in a 1-0 win against Germany to win his 32nd Wales cap.
But his managerial career began less than a year later when he was appointed by Fulham following Jean Tigana's dismissal.
He spent four years in charge at Craven Cottage, leading the club to their then highest Premier League finish of ninth, before being sacked in 2007.
Chris Coleman has been unveiled as the new Wales manager to replace the late Gary Speed.