Yes, for now bleach is best for your pool.
Since this is a new pool to you and not knowing how the previous owner cared for it, you can pretty much bet that they followed pool store advice and used a steady diet of trichlor pucks in a floater and dichlor shock powder regularly, both of which add CYA. If there is an inline chlorinator like a Pentair Rainbow 320 or something like it (looks like a big cylindrical tube sticking above the plumbing on the return side after the filter and heater), you should open it up (when the pump is off) and make sure there are no pucks in there either.
Well let's just say I am shocked that the pool is in the shape it is in and not worse, the house was a foreclosure, then flipped, the flippers did not do much with maintaining the pool. I am assuming it would have been green if the water temps were much higher.
What do you guys recommend for adding bleach, such as amount and type of bleach, and how often with the levels I am currently at? And there is no feeder.