I have read all solutions above and neither one worked for me because my main character had 1 Luck. So I figured I had to start a new game. I just did the Hardcore achievement in 2 hours thanks to this site so I figured I used that method to gain quick access to the casino's with high Luck. Here is what I did:
1) Start a new game
2) Skip the opening scene
3) Character appearance doesn't matter since this won't be a playthrough
4) In the S.P.E.C.I.A.L menu, make sure you pick Agility 10 and Luck 10, the rest doesn't really matter ( I personally picked Charisma 10 too for a higher Barter skill, this way you can sell the Combat Armor's for more caps, this makes sure you can start at the max bet. )
5) OPTIONAL: You can pick up the stimpacks and doctor's bag in case you run into trouble on the way to The Strip
6) You can pick whatever tag skills you like, I picked barter, speech and medicine.
7) Leave the Doc's house.
8) Travel EAST to Hidden Valley, watch out for Powder Ganger traps on the way, you can outrun them but you can get crippled easily by their mines ( I do not travel trough Sloan cause the sneaking is quite a hassle, this may be a longer route but you run pretty fast )
9) From Hidden valley, take the EASTERN entrance towards Scorpion Gulch
10) Continue travelling EAST and as soon as you hit Helios One, head NORTH towards the Durable Dunn's Sacked Caravan.
11) At Durable Dunn's Sacked Caravan, loot the 2 thugs from their Combat Armor and loot the other body for Leather Armor. Travel NORTH to the Crimson Caravan, sell the Combat Armor for 1500. Then exit through the WESTERN entrance and head SOUTH-WEST.
12) You will come across Gun Runners, sell your second Combat Armor for 1000. You will have roughly 2500 caps.
The next section can be done 2 ways. You can do the regular Credit Check at the Gate which is the original method or you can gain access via the Monorail in Camp McCarran. The Optional Route is described in the Notes section.
13) Enter Freeside through the EASTERN entrance. Go through the blue door next to the Mick and Ralph's crier.
14) Head towards The Strip and pass the credit check.
15) Now enter the casino of your choice, I started at Gomorrah.
16) Play Black-Jack, play with bets at max which is 200. If you have luck 10 this will go very fast, within half an hour I was kicked out of all three casino's.
17) When your cards are: 8-14, double down, 9 out of 10 times you will win
When your cards are: 15-16, hit and stay and win
When your cards are: 17-20, stay ( You can hit at 17 but that is quite risky )
The profit you have to make before you are kicked out of each casino is:
Gomorrah: 9000
Tops: 10000
Ultra-Luxe: 15000
NOTES:
As mentioned by Cra1gm0nkey in the comments, make sure the boss of the casino you are working on, speaks to you and kicks you out of the casino. This is needed for the achievement to pop.
As Segendary Ty pointed out in the comments, you can also gain access to The Strip via the Monorail in Camp McCarran. To get there, go SOUTH SOUTH WEST from Gun Runners past NCR Sharecropper Farms. Once you have entered Camp McCarran, go to the big terminal building. Once inside, go to the EAST side of the building and go up the broken down escalators. To the left, will be a door that leads to the Monorail. Since we haven't done any reputation quests, the guards will turn hostile when you enter. You can go out of your way and try to obtain NCR armor or recuit Boon, the sniper so you can enter without the guards turning hostile. You might survive, you might not. I haven't tested this and this was submitted by the community so if you want to save some caps which doesn't really matter, you can try this. It won't save that much time either.
This method took about an hour/ an hour and a half (Feedback given in the comments proves this ). Hopefully someone reads this and helps him/her. Good luck!
PS: If you give me a negative vote, please comment why so I can do something about that.
PSS: Sorry for my bad English, it isn't my native language.