Well, of course it happens some of the time that one or both of the players start an attack on the king, working with actual checkmate threats. For instance, in the Sicilian Dragon main lines where White castles long, both sides will try to open lines against the other's king and try to checkmate him.
But most games are decided in a more prosaic manner -- one of the sides wins material, trades down into an endgame, usually promotes a pawn and then checkmates the opponent. Usually the loser resigns some times before this whole process is carried out, when it's obvious that he can't do anything about it anymore. Even those Dragon games are quite likely to end with both sides avoiding checkmate, but one side still lost for another reason.
So it depends on the position, but in most chess games direct checkmate doesn't play a big role.
At what stage in the game should we start thinking of a checkmate stratergy? From the beginning, mid game or near the end when only a few pieces left? Or not at all and let the game progress to a natural end? I find that once I start thinking of going for a checkmate, my ability to control the rest of the game slips sometimes quite considerably.