I've been playing Master Yi for 3 years. I have played ranked in 2 out of 4 seasons, and successfully managed to hit Gold in both seasons. My goal is to share my tips and tricks on how to manage to climb all the way to Gold with Master Yi.

A general tip which can save you LP is by knowing which picks to fight. Sometimes being overconfident in your skills can cost you the game. Therefore be extremely cautious when picking a fight. If you know you're behind, try to catch up by either farming or by making sure to initiate in teamfights after your team has dealt their damage and when you're fully stacked.

A common mistake in low elo is the assumption that your team will back you up. Generally it's hard to follw up Master Yi, as he has increased movement speed once he uses his ult. Therefore don't expect your team to be by your side when initiating fights. Rather pick your fight wisely, and the best way to pull it off is by letting your teamates initiate.
Master Yi is terrible at engaging in teamfights as he only has his Alpha Strike and Meditate to somewhat prolong his life. Once that's used the enemy team can just kill you in an instant. Therefore coordinating with your team is important.

If your team doesn't engage and asks you to do so, then don't. Remember, going in alone is the stupidest thing a Master Yi can do as the likelihood of dying in low elo by initiating is high. Unless you're confident in your team that they can follw up, then maybe a fed team member could carry, but don't expect to come out alive.

The best way to engage as Master Yi if your team doesn't engage themselves in fights, is by Master Yi taking his opponents down one by one. In higher elo during mid-late game, the enemy team piles up and pushes a lane. In low elo, people tend to do solo objectives or randomly taking camps. Master Yi can easily use this to his advantage as he is strong at 1v1's once his Rageblade is complete.

Pros

Very strong at witherning down opponents by Auto Attacking.

Almost unstoppable once he completes his Guinsoo's Rageblade

Very good at kiting camps.

Great at cleaning objectives such as Rift Herald, turrets and dragons.

Good jungle sustain once meditate is learned.

His Alpha Strikecan follow up Flash

Can solo Baron Nashor once he has full build.

Cons

Squishy.

Vulnerable against stealth champions such as Vayne.

Bad at initiating teamfights.

Items such as [Zhonya's Hourglass]] and Stopwatch counters Master Yi.

Vulnerable to any form of CC.

Early Game

During the early game you want to farm. If there is an opening at the different lanes feel free to gank.
If you are being counterjungled, try to either look for kills or counterjungle the counterjungler. Iverntends to counterjungle. If Ivernis taking your Red Buff, go to his jungle and take his Red Buff. Whatever you do, you do not want to fall behind. Exp is crucial in the early game, if you fall behind in terms of exp, you will not be able to keep up, leading to you to be useless.

Master Yi is excellent at securing Rift Scuttler, therefore secure them. In lower elo, junglers contest the Rift Scuttler, and Master Yi shouldn't fear to contest it, unless the enemies are on the extreme danger list listed above.
Also, try to secure the Dragon as soon as possible, as it is a huge buff early game.

Mid Game

Once you complete your Guinsoo's Rageblade, you do objectives, ganks, 1v1's, get turret platings etc. You are at your peak. The enemy team has also achieved to purchase their powerspikes, but very little beats a Master Yiwith Guinsoo's Rageblade. The best way to exploit the Guinsoo's Ragebladeis by securing the Dragonand doing ganks, which helps your team to get ahead.

Late Game

Ideally by this time Master Yi should have completed his Guinsoo's Rageblade, Wit's End, Skirmisher's Sabre - Bloodrazor and Boots. You excel at taking opponents down because of your fast Auto Attacks. Therefore, late-game focus on getting kills, especially if they are 1v1s. Also, kills are guaranteed on enemy's who have used their CC skills or have no CC moves at all. Mobile characters such as Yasuo should be no problem, as Highlander makes Master Yi catch up.
During teamfights you want to initiate once your team has initiated. You could initiate if the enemy team has no hard CC or if you're good at dodging CC moves by Alpha Strike or if you have Quicksilver Sash. If your Guinsoo's Ragebladeis fully stacked, you should be able to kill any opponent who is not a tank. Tanks take a longer time to kill, making you vulnerable, because as Master Yi you want to kill the damage dealers as soon as possible. Therefore focus on the damage dealers such as the ADC's as Master Yi .

Master Yi is an Auto Attack based character. Master Yi's fighting in the early game is alright. You shouldn't be confident in your damage as Master Yi in the early game, unless you're fighting characters such as Ammumu, Skarner, Ornn, Lee Sin which are all weak without their items.

One trick is the Meditate reset. After an Auto Attack, instantly Meditate to faster reset your Auto Attack animation, thus you deal an extra chunk of damage. This trick is super helpful.

During a fight you want to time your Alpha Strike to dodge skillshots and turret shots. This can be done on any targetable enemy champion, and any minion, thus being super versatile. This trick can help you survive tower dives and worries your opponents. This bug makes you untargetable, but not invulnerable, so Igniteor Kalista's Rend, cannot be dodged. Still, this skill is extremely useful, and will definitely secure you some kills without dying.

Finally, stacks. Master Yi's best fighting tip is to stack his Guinsoo's Rageblade. Initiating with an unstacked Guinsoo's Rageblade makes your attacks really weak. Therefore Auto Attack anything on your way to a fight, or speed it up by meditate resetting.

Just remember to play smart. If you play smart in low elo, you will succeed with any champion.