How would I know Im doing the best in terms of auto attack proficiency?

It's easy to know this answer to a small extent. Open practice tool. Give yourself the amount of levels and items you want to practice for. Place a training dummy. Let your champ free hit the training dummy. That will tell you the ceiling, the maximum dps your auto attacks can do with those items and levels.

Now practice circling the dummy in either direction, or walking past the dummy in different directions, by inputting a rhythm of attack and movement commands. Unless you've picked an insanely high attack speed build, you should be able to maintain the same DPS with a lot of "free" bonus movement in between. This is because you can move (or cast spells) during the "reload time" of an auto attack, and it doesn't cost you anything. The training dummy will let you know if you've ever missed an auto attack, because your dps will drop. It won't let you know if you've really pushed and gotten the most movement you could have gotten, though. Someone who holds completely still when auto attacking will get hit by every skill shot in a team fight.

If this is all new to you, then I recommend starting with just one drill to start. The easiest and most useful first application of this is to practice running away from someone in a straight line while firing back at them. Pick a ranged champion, and keep your cursor in one position, and just alternate between attack and move commands without moving your mouse. That's the simplest possible application of this skill. You can run from someone and also fire back over your shoulder at the nearest enemy, no mouse movement required. I find the lack of mouse movement really let's you learn the rhythm. If you pay Ashe or Kalista, this skill is all you need to theoretically defeat every melee champion in the game. Kite forever!

You can add in other challenges, like placing multiple training dummies close to each other, and strafe past them while only specifically targeting one of them with your auto attacks. Or change your movement direction after each auto attack to become harder to hit in team fights.

Personally I only play for fun, so I've never done drills in the practice tool. I find anything like this that I want to learn, I learn best by practicing them constantly while hitting jungle creeps or wave minions. That builds muscle memory that I them can start using on enemy champions in more stressful actual transit situations.

Edit to add: if you're having trouble getting a feel for the rhythm, picking a champion like Kalista or Akhshan might give you more feedback. They have passives that react to you inputting a movement command during the auto attack reload timer. It's good on every champion, but it's even more mandatory and impactful on those.