How do you get the Season 2016 - 5v5 Platinum?
It was a ranked reward. Riot's description in the client reads: “Earned by achieving Platinum in the 2016 Ranked Season in 5v5.”
Is the Season 2016 - 5v5 Platinum still available?
No. The Season 2016 - 5v5 Platinum was a ranked reward: to get it you had to reach Platinum in Ranked 5s during Season 6 (2016). Riot does not give out past ranked rewards again, so it cannot be earned anymore.
