What is the best strategy for quickly getting Badass tokens?

Solution 1:

• Switch up weapons - you get challenges for number of kills with each type.
• Switch up elementals - again challenges for number of kills.
• Switch up grenade types - there are challenges for using each one.
• Switch up shield types - there are challenges for kills with fire novas, 'roid kills, etc.
• Each map has a set of challenges, beat bosses in a particular way or find hidden symbols and the like.
• You can follow a particular challenge if you're close to completing it, which bumps it to the top of the challenge list.
• As you progress the challenges get harder, but have greater rewards.

Solution 2:

Assuming that you have a friend who is willing to assist you with it, and assuming you do not mind a tactic that is considered somewhat "degenerate", you can quickly rack up large amounts of Badass Ranks by repeatedly completing the Level 5 "Psst, Hey, Buddy..." challenge. This requires trading with another player a total of 50 times, but no loot or money needs to pass hands - merely the act of initiating and finalizing an empty trade will give you credit toward the challenge. Each new character pair that you and your friend create and perform the 50 trades with will earn you a total of 166 Badass Ranks, which apply to all your characters. Characters used for this can then be deleted and recreated to harvest further Badass Ranks.

Edit: It should be noted that one of the PC/Steam patches for Borderlands 2 may have removed this trick. Those with the Gearbox guns due to Premiere Club membership can use these to have something to trade.