I am wondering if I need the starter pack to play this game. Are there any pieces of it that are essential, or will I find suitable replacements in the booster packs? I want to run a draft tournament from some booster packs, but we'd prefer not to buy a bunch of starter decks.

There are no Story cards to be found in the Asylum Packs. There are Conspiracies in the APs, not quite the same thing. Core Set and the upcoming Secrets of Arkham Expansion are the only ones with Story cards to my knowledge.

Joeyman said:

Story cards tie directly into most of the winconditions of the game (you'll need to win 3 story cards to win the game, most of the time) so, they are quite essential. Tokens etc. are easily replaced by coins or beads, and in the CCG era we used just that.

You mention 'boosters' which could indicate you localized some CCG era product. Boosters are random products, while in the LCG the cards are only available in a fixed format. If you found CCG stock, you may have found one of the old starters, which include a story deck.

If you didn't find CCG stock (and probably if you did) the Core Set is a good, cost effective way to get started, as it has everything you'll need and a good selection of cards that fit the basic functions a deck needs.

If you somehow own a collection of cards, but not a story deck, the upcoming Secrets of Arkham Expansion will be a good place to start playing, as it'll contain the official tourney story deck for 2010.