I have read some reviews on AGoT LCG that have made me interested in it.  I have not yet read the books that the game is based on (and know that it is not necessary). 

I was wondering how much a person would need to invest in to get multiple decks of the same faction that have different feels to them.  How different of a feel can you get from the same faction?

With one copy of the Core Set ($30) and one copy of your favorite Deluxe Expansion ($25), you would be able to build multiple decks for the same House, and they would have distinctive "feels".

 The Core Set and Expansion boxes are made to go together, the expansions provide two deck lists that you can build using just those cards... that said you could not build both decks simultaneously (that would usually require at least one more Core Set).

If you want to have two good decks with very different feels I'd recommend the Core Set, the Deluxe Expansion, and then one cycle of Chapter Packs. If you only ever want to play that House you can use the extra cards from the other Houses as trade bait for copies of cards you really want more of.