To get the achievement The House Always Wins you will have to complete all quests given by Mr.House, to start getting missions from Mr.House you will have to enter the New Vegas Strip. To enter The Strip you will have to get a passport the three main ways of getting one of these are, 1. Pay the robot at the entrance 2000 caps, 2. Go to Mick and Ralphs and convince Ralph to sell you a passport for 400 caps or 3. Complete all quests given from The King in the G.I Blues side quest (This will also earn you a achievement).Also pointed out by MetalThreshkeen if you are liked enough by the NCR you will be able to enter The Strip by the train in Camp Mccaren or as pointed out by CRAZYdaan if you feel like you have good enough weapons, try and kill the robots protecting the entrance. Once you have entered a robot should come up to you and tell you to go and see Mr.House in Lucky 38 Casino.
After you complete the quest Ring-a-Ding-Ding given by Mr.House (Also a achievement gained for this) he will give you the set of missions that will unlock the achievement these are;
1.The House Always Wins, I
2.The House Always Wins, II
3.The House Always Wins, III
4.The House Always Wins, IV
5.The House Always Wins, V
6.The House Always Wins, VI
7.The House Always Wins, VII
You will unlock the achievement when you complete the final quest for Mr.House and agree to go to Hoover Dam.
If you having trouble with these quests go to this link http://fallout.wikia.com/wiki/The_House_Always_Wins
All credit goes to Fallout Wikia for the detailed walk through.