How do you analyze League of legends players' gameplay?

To analyze gameplay, I focus on the intention of the player in the moment, what they are aware of and trying to accomplish with their play, and then compare that with what is the ideal play for the moment based on the info that is available to them in that moment. Often they won't gather all the relevant information, such as missing the jungler pathing up to them, which is negative and can ruin the play, but they could be focusing on the wave and the dive and reset timers correctly, which are all perfect. Looking at this play as wrong because the result was wrong isn't usually helpful.

Instead, calling out all of the correct aspects of the play, and then adding in the need to track and learn the piece that was missed to improve, is the best way to analyze and improve. This is especially true because it helps the player feel confident and seen, while also showing them what is usually a consistent flaw in their gameplay and game sense, that when they fix it, will give them 3-5%+ WR.

That's a simple question, I put the game under 3 phases. 1- Early Game - 2 Mid Game 3- Late Game. then we start understanding the pros and cons of the player in every phase. and every single phase is also expanded into multiple categories such as 1- CS 2- Trades 3- Wave Control for early game.
1- Roaming 2- Objectives & Rotations for mid game.
1- 5v5 decisions " Whether if you should scale and chill or try to rush objectives."
2- 5v5 team fight positioning and obviously late game itemization. which is also related to early-mid game anyway but yea. 3- simple macro to try and close the game as fast as you can. such as trying to split push or ignore objectives to finish off the game instead while enemy team's having a BBQ party on the baron. sometimes late game is pretty random so yea.. you just have to adapt and that's what I try to teach the player.

Again, it heavily depends. I won't go in details of waves core mechanics if a player doesn't even know how to punish opponent that had locked himself in the auto-attack animation. We probably will never talk about that one Baron fight that with the same setup most likely will not happen again (things where we have to look at X1, X2, X3, X4, "player, Y1, Y2, Y3, Y4, Y5 where X is allies and Y is opponents; looking at all the levels and/or items factors; mana; cooldowns; wave states; game state; etc.) and definitely we shouldn't opt into this if one's laning phase is far from perfect.
Most important thing in gameplay analysis is the coaches ability to select, focus and outline parts that actually matter for the coachee (student).

Depends on various factors. I always start with analyzing the student's account (roles, champs, rank, winrate, kda, ...) and ask questions to find out more about my student's mentality, discipline, approach, ... .

Typically for my students in the lower elo bracket (everything below D1) I start with viewing their gameplay from a 'fundemantal scope perspective'. I look at their gameplay and specifially analyze how well they do on the fundamentals and then from there on out I build up the session.

As for my higher elo students, I focus a lot more on on-the-spot (real-time) decision making as this typically tends to be the main struggle for higher elo players.

Of course I will always adapt things to my student.

Before the coaching session, I ask my students if they have a clue what they are doing wrong in the game. It helps me to get to know them better and I am more aware of their strong and weak sides.

When I'm watching students in-game I always pay attention to their advantages and disadvantages. If they are doing something properly I'll let them know but when they are doing something wrong I explain to them what is the problem and how to fix that.

For accurate analysis, I need to do a VOD review with my students. During that, I give them pro tips from my experience as a player and coach, and we talk about mistakes. I propose a solution and I leave my student with a training plan.

My preferred method is the VOD review/ watching a replay of a game recently played. This way we can both have a look over that game, the coached person can explain to me what went to their heads in that moment, and see my analysis in the present, while I take timestamped notes that I further send to the coached person after we’re done.

The secondary method is live game coaching: this allows me to see how they watch the map, how they think about trades, how they approach fights, and actively have a dialogue with them before they commit to a decision, so they can have a little helper that sets them on the right path.

Gamestate is king. Analyze what's the current and past state of the game to determine best possible outcomes. After that, analyze whether, if the correct outcome was chosen, was the execution also correct.

Don't ignore mechanics and understand the real issue behind each thing. For example, be good at analyzing why you got hit by a skillshot, rather than saying that getting hit by it is bad/"just dodge". Analyze why it wasn't dodged down to the mouse usage. Avoid things like "you were greedy", instead giving a proper analysis of the risk/reward

It is about breaking it down into small parts.
Usually I would start from the little things. Subtle movements, minor habits. And from there I would then try to look for patterns, and then figuring traits and playstyle out of a certain player.
With this method, I am capable to understand the strengths and weaknesses for my students. On top of that, I also get to understand what is the cause and the root which leads to them having the current playstyle.

When I watch a live game I usually watch movements camera, hotkeys what the player focuses on and some minor macro things just to avoid that the game gets lost even before the start. If i'm watching a replay i'm more focused towards gold efficiency reset, matchups, trading patterns, warding and so many other things. Once I realized which one the customer does worse I'll do the first lesson on that specific argument!