Instructions
Complete the entire "Dead Money" storyline. By the end of the main storyline in the Sierra Nevada region, you will have obtained at least 4,000 experience points, which is plenty to upgrade your character even at a high level. Of course, this experience-point count is not including the small bonuses granted for unlocking doors, passing speech checks, and so on.
Complete any side quests you encounter. The map in "Fallout: New Vegas" is filled with characters and situations that will each begin one of the game's dozens of missions. Once you have completed the "Dead Money" quests, you will still have open access to many of the game's other side quests. In general, quests are worth far more experience points than any individual tasks or enemies. If you find yourself stuck, talk to everyone in the city of New Vegas to find new quests.
Search for regions of the map with high-powered enemies, because these monsters are worth large amounts of experience points for your character. Notable locations include the Quarry, which is located north of Sloan and filled with difficult Deathclaws, as well as the Cazador nest, southwest of the Hoover Dam.
Return to older dungeons and vaults. Complete any lock-picking or science puzzles you may have been unable to complete previously. The experience-point benefits of these tasks add up quickly, allowing you to potentially add levels without encountering high-level enemies or quests.