No class really changes your race, but some change your creature type.
Usually such creature type changes are capstone abilities (i.e. gained at level 20), like the Monk's / Unchained Monk's Perfect Self, turning the character into an Outsider. There are several such capstones, sometimes as such plain class features, sometimes with specific class paths and sometimes as an archetype feature (e.g. Planar Oracles).
Some creature types are more common – like the Outsider – while others are very rare, like the Nature Oracle's ability to turn into an Animal or Plant creature.
Changing your race is still possible, though usually with spells like Reincarnate, instead of with class abilities.
Humanoids are also considered to be of a specific race, if they have a racial subtype, so acquiring such a subtype would add a race to your character:
Advanced Race Guide:
If a humanoid has a racial subtype, it is considered a member of that
race in the case of race prerequisites.
For example, a humanoid Sorcerer, who has the Orc bloodline, gains the Orc subtype and henceforth counts as an Orc.
The Racial Heritage & Planar Heritage feats treat your character as if it belonged to a specific race, though you keep your original race in this case as well. The Mythic version of Racial Heritage grants you one of the chosen race's racial traits.
There are also ways to become Undead, for example turning into a Lich or Vampire, though these don't rely on class abilities either: Occult Rituals like Eternal Apotheosis or Fourth Apotheosis provide everyone with the ability to turn into a Lich or Fiend (evil Outsider) respectively. Druids can turn into an undead Siabrae with Welcome the Blighted Soul.