On this page of the game guide to Borderlands 2 you will find a detailed description of the side mission titled Breaking the Bank. You can begin this quest in the Lynchwood location during the completion of the side mission named The Bane.
1) Bounty Board
3) The building from which you can access Laxative
5) Mad Dog
7) Hideout 1
8) Hideout 2
9) Hideout 3
Approach the Bounty Board  and accept the mission. Find the bank  and try opening the door from the above screenshot.
On the roof of the building opposite to the bank you will find the Laxative. In order to get there, jump there from the roof of the building marked as . The item itself can be found in the toilet.
The next part you need are explosives which can get them by killing Mad Dog . In order to reach him, use the elevator .
Go through the cave . Place the explosive in the designated spot and pour Lexative onto it. Soon afterwards a Skag will head out of the cave and eat it. Follow the Skag and check whatever it "returns" to find the bomb.
Now return to the bank  and plant the explosive on the door. Detonate it and you will be attacked by enemies. With the door destroyed, head inside and search all the stashes (25 in total).
After robbing the bank, you will have to hide your loot. The first place can be found below the conveyor belt . The second is inside the cave  and the last one behind a piece of steel from the above screenshot. Now return to the Board  to finish the mission.