Yes it is, but you need to keep your 360 in order to do so. You basically fire up the game on 360, make sure it’s fully up to date, and make sure you have your Zer0 character loaded at the main game menu screen. The bottom menu item lets you upload your current (default) character. This also uploads heads, skins, BAR ranks, etc. Then, you fire up your XB1, fire up the game, go to the same menu and download the character. If you have multiple characters, you have to do them one at a time.
Edit to add: even if your 360 saves are cloud-based, you still need to transfer one at a time using the in-game menus.