This question might appear broad, however there should only be a few general methods for farming caps, disregarding glitches and exploits.
Exploring the Wasteland
Your dwellers will find caps in the wasteland, but they will also find weapons and armor. These items can also be sold for caps.
A dweller will find rarer items, worth more caps, the longer they are exploring. The dwellers Luck stat plays a part, too. A known method for fast forwarding this process is to simply send them out, and fast forward your phones clock. If you set the clock a day forward, explorers will immediately have a day added to their exploration.
It might be worth considering the bounty they hold in comparison to resurrection cost, in the case of dead dwellers, but please be aware that altering the clock can have other adverse reactions, as already seen from some of the other questions being asked about Fallout Shelter at present.
Leveling Dwellers and Collecting Resources
Every time a dweller levels, or you collect resources from a room, you are awarded a small sum of caps. A rooms cap reward depends on the collective sum of luck between its workers, while a dweller yields an amount of caps equal to their new level, upon leveling up.
Rushing a room gives you a caps bonus. You have a chance to fail, but surviving any repercussion tends to yield a small amount of caps, and experience for all dwellers in the room. If you keep an eye on your dwellers health values, you could easily rush a room repeatedly for greater reward.
Note : The cap reward is equal to the chance of failure, and is independent of the room size or occupancy. Thus, having several small, unupgraded rooms can be used to quickly farm caps, without the more dangerous incidents of larger rooms.
Add high LUCK dwellers to resource rooms
Every time a resource is collected, there is a chance of getting extra caps based on the workers' LUCK stat. By having many resource rooms filled with dwellers with high LUCK, you will gain a steady income from just collecting resources.
Lunchboxes often have cards of 100 or 500 caps inside them. You can keep pushing objectives to unlock the odd lunchbox reward, or simply purchase them for real money.
There was a known exploit for quickly generating "free" lunchboxes in the first iOS version of the game, but this was patched in the first update to the iOS version and was never available on the android version.