League of Legends (LoL) Question: How can I farm under tower more eficiently

There are several different things you should learn to farm under tower effectively.

For starters, and as said in the other response, you need to know how many turret shots it takes for every minion to die. In an ideal scenario, a full hp wave will mean the melee minions take two turret shots then have very low health, while caster minions take one turret shot and have about two auto attacks worth of health. You can more often than not just onehit the melee minions after two shots (if you wait until the third turret shot is launched you get more time to position), while caster minions will need to be hit once before the turret hits or most champions won't be able to farm them. The cannon minion takes long enough that you can click it to make a rough estimate of when it will take it one turret hit to die and either autoattack it or use a skill if its health is too high. Obviously, all of this gets more complicated when your minions are attacking the enemy minions, so you gotta use your own judgement to try to make them be low enough to be farmed right after a turret hit.

Other than that, you should know that some champions will make it harder for you to farm, especially if they are ranged and can damage you under tower. You'll have to make sure you can do the turret hit math subconsciously because you should be focusing on the enemy. The point is that if they hit you in the wrong place and moment, they will get a turret shot and will most likely lose the trade. Use your best judgement to know whether it's worth to do the trade or not.

Also, as a last point to make, it's better to lose farm than die and feed your enemy. There's not much you will be able to do in a lot of situations, so don't be afraid to just back from the turret if you're too low on health, because otherwise you will lose both the gold and the experience from the minions on top of giving up a kill.

In early game it works like that
for high basic attack damage
for melee minion - 2 tower shots and then 1 basic attack
for caster minion - 1 basic attack, 1 tower shot and then 1 basic attack.

for low basic attack champions like mages you need one more basic attack for melee, do it before the first shot or before the second shot so you will be able to land the last basic attack.

keep in mind minions does damage too, so try to see which one they focus.

better to attack sooner the later with minions, and if you are range look for the basic attack travel time too.