Personally, I have attack move bound to a button on my mouse. That's the easiest in my opinion, but not everybody has extra buttons.
You'll want to make sure you have the setting enabled where attack move attacks the thing closest to your cursor, rather than the thing closest to you. That way, you get a lot more control over what you're attacking.
Basically, the idea is to "stutter step". You hit your attack move button to manually attack, then click your move button to make a small movement between attacks. It's gonna require some rapid mouse movements to get it right, clicking between your target and the direction you want to move, but the result is that you slowly move around while dishing out damage, making you more of a threat.
If you really need to, there is also a toggle for "Target champions only" that you can turn on for teamfights or when the enemy is trying to use minions/wards/towers to prevent you from clicking on them. That way, attack move won't accidentally hit minions.