I have two boxes of booster packs (the old ones) and a bunch of the tournament packs. Is that enough to play?
Since you're talking boosters, you have the CCG. A lot of the CCG cards are unplayable in the LCG format. You might have to search for the LCG cards on a site like CardGameDB.com to see which cards you have that carry over. Only cards that 100% match an LCG version of the card are playable in the LCG.
As for needing a core set, the answer is yes unless you have someone willing to give you a few cards. Most decks will need Valar Morghulis or Wildfire Assault. You could get away without buying a core set and just buying an expansion box if you did that. Again, this is all referring to the LCG. The CCG is defunct as far as official FFG support.