An easy way to sum up the basic process I use with clients is EAL.
Evaluate Movement, then Achieve Good Movement, then Load Good Movement.
The basic cycle works like so:
For any given movement that a client does, evaluate it. Is it good, functional, balance movement? Does it cause pain, reveal imbalances, or otherwise function poorly? An example is a bum knee - is it a bad knee or the symptom of a hip or ankle problem? Is the weight more on one side or the other? Does the person's gait or squatting or pushing or pulling reveal a series of compensation patterns that are eventually going to lead to the former two issues?
Evaluation can be a formal process, but it should be an ongoing process. Watch everything your clients do and ask, is that good movement? If not, what could be causing that? Test and evaluate.
Achieve Good Movement
Once you know what's holding the person back, you have to improve it. Break down movements they can't do into individual bits you can correct. If someone can't squat without rounding at the spine, use your evaluation process to find out why. Then use movements they can do, or ranges of motion they can achieve, to start to correct the issue. Weak rear deltoids causing problems with pulling and pressing? Use direct training on those rear deltoids to bring them up to speed. Knees collapse in because of poor strength in the quadriceps and hips? Band walks, slow static lunges, isometric holds, and squatting with bands might help improve this. And so on.
This stage also involves teaching. It's not always a weakness that drives poor movement - it's just not understanding what proper movement is. If you've been taught that your spine moves like a slinky in a pushup and that's okay, then you may have weakness but you've also been allowed to train in bad movement. Teaching the client occurs here - showing how to move in a way that's more efficient for life and/or sport.
Load Good Movement
You improve strength and musculature with load. But load works counter to good movement. It's a challenge to move well under a heavier load than you moved under previously. That's a normal and expected part of the process. You will hit your limit of your ability to do the movement correctly (aka technical failure) before you hit your limit of your strength to do the movement.
Load gets added once you get someone moving well. It will reveal limitations, and it may reveal poor patterns concealed by overcompensating - bad back position on a deadlift that's okay under low weights because your back can take it but not under heavy ones. Squats that look okay without weight but which collapse under your first significant load. And so on. You can't fix a bad movement by loading it. You fix a bad movement as above - by directly addressing the issues by training the movements and ROMs you can do and improving them, and by imparting proper movement and form. Then you load it.
I call this EAL cycle a cycle because it's iterative. You keep doing it. Evaluate, Achieve, Load, and Evaluate again. When movement goes from "good" to "bad" you apply the cycle again to find your next step.
This is not a very basically worded post, but this is an underlying foundation - a true basis - for how I train clients.