First, make sure that your game is updated with the latest version available : heads are available on mobile since update 2.5.6 (February 18, 2025).
Once it's done, go say hi to the Tailor's Daughter, in Prisoner's Quarters. At start, only the Classic Head, the Violet Hole, the Vortex and the Blowtorch are available but you can also find the Bobby Head in a lore room in Prisoners' Quarters (some heads unlock retroactively if their goal is already met on an existing savefile).
For the other ones though... You'll have to fight ! Locked heads offer a hint on how to unlock them.
You can see all the available heads, and how to unlock them, on this page: https://deadcells.wiki.gg/wiki/Heads
Beware, some Outfits that already come with a custom head can't have their head changed:
- Reverse Burglar's Outfit
- Winter Outfit
- Cultist Outfit
- Robber Outfit
- All crossover Outfits, except for Vessel's Outfit
- All non-boss Return to Castlevania DLC Outfits
- All Death Outfits
- All Scarecrow Outfits
- All Servant Outfits
- King Outfit
- White King Outfit
Please note that checking "Multiple deaths" in Assist mode prevents unlocking heads related to dying, even if you choose to end the run and not revive.