First of all, you have to know when to play an Assassin. If the enemy team is mainly made of Fighters, your assassin won't live long enough to get much done. You should play assassins when the enemy team has 2-3 soft champions that do enough damage to be very threatening.
In the current style (heavy emphasis on Fighters), assassins are risky picks, and your team should be very careful about the choice. As it is, though, assassins are still really great to have on the team when the conditions are right. The best games I've played lately have been as assassins.
Again, the most important thing is making sure an assassin is the right choice for the team you have and the team you're playing against.
Opening the door for an assassin is the job of your team, you just have to be ready at the right moment. Even the best players will fall to assassins if the timing is right. Too many assassin players try to initiate fights, which is a mistake.