No, you can't adjust PH and alkalinity with chlorine (bleach), but the chemicals to adjust ph and alkalinity are common household items (baking soda, 20/20 mule team) and muriatic acid which one might have in the garage. PH down is dry acid, an alternative to muriatic acid but a little more expensive.
The point really is that out of the chemicals the pool store sells you, the only ones you actually need are the exact same as you find in the grocer and hardware store. All the other stuff, the algaecide and sparkly pool goo of various sorts are unnecessary.
The other point is, once you get your water balanced all you will need daily is chlorine. Some pools have fast rising ph, so a weekly dose of muriatic acid is also added to the soup. But many of us only have to add bleach all season once the water is balanced. There is the occasional ph drift to adjust, and maybe if it rains a lot or something gets into the pool, or a lot of water is drained/added will there be the need to re-adjust.
Knowing and targeting an appropriate CYA level for your pool is where you start, effectively setting the pool at a certain platform from which you can control the chlorine levels easily. Without CYA, ALL your chlorine will burn off by sun and bathers every day. Your pool will drop to 0, and that is not good because things will grow. With a lot of CYA, most of the chlorine in the pool is inactive and bound up under sunscreen which means you have to make sure it's all there under the sunscreen, plus add enough atop that level to actively disinfect. If you drop below the inactive level, it starts using all that inactive chlorine and will eat it up fast. And this is when again, things start to grow.
At a moderate CYA level (let's say 40ppm), your inactive chlorine is in the 1-3ppm range. Always there, always inactive. From 3-7ppm the chlorine is active, killing stuff, and allowed to burn off via sun. Every day you re-dose up to 7ppm, sleep, wake, swim, sunbathe... and end up at the end of the day still above the inactive level. This allows you to never fall below minimum (3ppm), and always have chlorine in the pool no matter if you use all your active chlorine every day, or not.
CYA is your sunscreen, disinfection safety net, and protector for your chlorine. And like any good thing, too much is trouble.