For a frame of reference, I put about 2 gallons of liquid shock in my pool per week, and have a floater with chlorine tablets in it that I refill as needed. So don't fear, you won't need to spend that much on bleach once you have this pool under control! I have never put bleach in the pool so am not sure how that compares to liquid shock.
Mod note: This is not how TFP recommends killing algae or clearing up a pool. Here is the process we recommend, SLAM, pooldv