After Don Vito's death, the Corleone look weak. Tessio wants to make a deal with Michael, just as Don foreseen: the man that comes with that offer is a traitor...
Tessio - making future plans connected with the Family weakness
Get into the Embassy club. Just after a few seconds, a firing starts. You've got to kill Tessio - at the beginning he's in the basement, but later he will start escaping by the stairs. Use the elevator, then follow Tessio - a blue blip on the mini-map. Take care of two gangsters in the first room. There's a healing drink on the table. Just after the stairs leading to the ground floor you'll run into another two enemies. Finally, you'll catch Tessio in the bar room. You'll have a chance to talk to him, but it's pointless. Just shoot him to end the mission.
Just after leaving the building, talk to Tom. He'll tell you that Michael wants to see you tomorrow, before the baptism. Get into the car and drive to a the church in Little Italy.