As the achievement description advises you need to win every event in the tournament to unlock the Platinum Medal.
When I first started this game I found the AI to be a bit tricky on the Arena style events, you may not, but this solution is based on how I found the game.
To me it was far easier to win the Race based events, I used any collected cash to upgrade the Engine and Boost first before other Board options as I found this let me build a lead and I could go on to win.
I chose the 6 event Race based Tournament and was able to win the first 5 easily.
The 6th event is a Chaos event, I found that if I avoided the other riders and focussed on destroying Weapon Targets & Flags (circles & triangles on the mini map) I could rack up points without getting taken down.
On later tournaments I found the tunnel section to be a very good place as I was on my own, both targets re-spawn quite quickly and I could destroy them and rack up the points without getting taken out by the other riders.
You may prefer to stick near the other riders and earn takedown points that way, it's an option and one I used a fair bit in the later tournaments, but the 4 points for a Weapon Targets can rack up quite quickly.
As a side note, after I unlocked the Platinum Medal on this event, I replayed the 7 event Arena based tournament a second time and as my skills had improved was able to get a Platinum on that one as well. Focussing on upgrading the weapons over the board was key, in particular the blue mines.