Checklists in Trello are a bit weird. A checklist belongs to a board. The same checklist can be on several cards. But each card has separate checklist item states (checked/unchecked). I don't think it's possible to add the same checklist to more than one card on trello.com, but that's how it's modeled behind the scenes.
As of release 0.5.5-beta Trello.NET supports this. Use Trello.Cards.ChangeCheckItemState
. More info here.
Edit:
This example shows how to loop through a card´s check lists and check items.
foreach (var checkList in trello.Cards.WithId("a card id").Checklists)
{
Console.WriteLine(checkList.Name);
foreach (var checkItem in checkList.CheckItems)
Console.WriteLine("\t{0}: {1}", checkItem.Name, checkItem.Checked);
}