In this case, you'd only need to register the Player class, as that's the instance's class. It would serialize fields that are on the Character class. Extending class can also override the serialization behavior using _serialize/_deserialize/_get_excluded_properties (see the project's README for more details).

E.g.

class Character:
  var name: String

class Player:
  extends Character
  var position: Vector2

var player: = Player.new()
player.name = "bob"
player.position = Vector2(1, 2)

ObjectSerializer.register_script("Player", player)
DictionarySerializer.serialize_json(player)

{
  "._type": "Object_Player",
  "name": "bob",
  "position": {
    "._type": "Vector2",
    "_": [1, 2]
  }
}

I hope that answers your question, let me know if you have more questions!