There are many ways to do it.

One very easy way is to use trichlor, with a floater or two, to maintain a constant chlorine feed, and bleach to kick it up 1x per week. By using both, your CYA will move upwards pretty slowly.

You know about bleach; here are the other items:

Trichlor tabs: buy at at Sams Club IF you can buy the PoolBrand product in the 40lb bucket OR Kem-Tek 3" Trichlor 35lbs @ Amazon . Do NOT buy from Costco, Walmart or local pool stores - most are selling trichlor blends which can be a real problem. If you find the Kem-Tek brand at Home Depot or Lowes, that's OK. Other brands, you have to learn to read the chemical ingredients section of the labels VERY carefully!

Trichlor floater: This one is typical, and fine - Kem-Tek 879 Pool Floating Tablet Dispenser Pool and Spa Accessories, but if you want to buy that locally, I think most of them are fine. If one DOES spill trichlor tabs in your pool, get them up quickly -- they will often leave a very difficult to remove stain, if left on the pool for hours.