How does true vault hunter mode work? Is it free? Can I play through a new game with a new character on true vault hunter mode? I have tried true vault hunter mode so I do not know about it.
True Vault Hunter Mode is like "New Game+" for Borderlands 2. It unlocks once you've completed the main quest in "Normal" mode. Then you can play it, for free, at any time you want, with any character that has completed the main quest in "Normal" mode. It's part of the base game, no DLC or anything else is required.
There's also "Ultimate Vault Hunter Mode" which is part of the "Ultimate Vault Hunter Upgrade Pack DLC" and included with the Season Pass. It adds a third playthrough after you complete True Vault Hunter Mode and are at least level 50.
In both cases (True and Ultimate), when you start the game your missions reset to the beginning. The difficulty and levels of the enemies increase. You can switch between modes at any time at the main menu. When you select a character, you'll get a prompt for which mode you wish to play in. Quest progress is tracked for each mode separately, but your character stays the same regardless of the mode.