Django Rest Framework provides a pre_save() method (in generic views & mixins) which you can override.
class NoteSerializer(serializers.ModelSerializer):
owner = serializers.Field(source='owner.username') # Make sure owner is associated with the User model in your models.py
Then something like this in your view class:
def pre_save(self, obj):
obj.owner = self.request.user
REFERENCES
https://github.com/tomchristie/django-rest-framework/issues/409#issuecomment-10428031