Well, she knows she's only a novice and of course she doesn't think she can do a bit of damage to a Dark God... however, she's a demon hunter and has vowed to dedicate her life to bringing the Light and destroying the darkness however she may. It's her life's calling and while she can only do a little, she'll do what's her due to undo evil. 

As for the independent part, well it's something I liked about 5e Paladins... their power doesn't necessarily come from the worship of a specific deity, instead it's drawn from their Oath and their conviction and dedication to a good cause.