Guide on Getting Good Placement Results next Season
Summary: Increase your Match Maker Rating (MMR) during Preseason by playing ranked and winning the majority of your games.
Long Version:
As soon as the ranked season ends for the purpose of rewards, your ranking and LP ceases to matter. The only thing that will carry over is your MMR which will undergo a 'soft' reset at then beginning of the next season. The rank you receive at the end of placements is determined entirely by your MMR after changes from the reset and the games you play during placements.
Placements on an account that has been ranked have much less of an effect on MMR than placements on an account that has never been ranked. As such you will be placed at about the level that matches your MMR from the previous season and preseason. The soft reset usually results in MMR being lower. This means even if you win most or all of your placements you will not have a meaningfully higher MMR after placements than you did before the reset.
MMR goes up when you win and down when you lose. If you have been winning a vast majority of your recent games, then it will go up more on a win and decrease less when you lose. The reverse is true if you have been losing a lot lately. Dodging does not affect your MMR.
I ended Season 4 with really bad MMR so even through I was Bronze 1 at the end went 8 wins 2 losses in placements in Season 5 I was put in Bronze V; however, I did really when in Season 5 so I had basically Silver V MMR at the end but due to internet issues for the majority of the season only played enough to get up to Bronze III. But this season after doing 8 and 2 in placements again I got Bronze 1 purely off of having ended the previous season with MMR much higher than my rank.
Don't be one of the whiny people posting about winning 7+ games in placements yet ending up with the same or worse rank than you have at the end of this season; if you actually care then make sure your MMR is sufficiently high enough at the end of preseason that you would have to lose all 10 placements to not end up with a higher rank than you currently have.
Pay close attention to the bold text.