This achievement requires that you earn a Platinum medal and will be the most time consuming and "grindy" achievement in the game. Before we get into it, just note that this does not require playing with your team in any way, and is actually easier if done by yourself. Each player has a rank that starts at Bronze medal 1. You then progress all the way to Bronze medal 99. Your next rank up will take you to Silver medal 1. Continue leveling through all of the Silver medals to reach Gold, and all of the Gold medals to reach Platinum medal 1, at which point this achievement will unlock.
The only way to level up is by playing and completing a "New Crossword" (one that you have never completed before and is labeled as such). In the default game, there is only around 20 unique puzzles to complete. To reach Platinum, you will need to complete 300 unique puzzles. The easiest and least time consuming way to go about doing this is to abuse Public games by joining New Crosswords that other players are hosting. From the Public puzzle selection screen, constantly be selecting New Crosswords (you are allowed to have up to 5 running at a time). Simply joining the game and then backing out will be enough to get credit. Once the puzzle is completed (it will say 100%), click on it in the Continue Puzzle screen to be awarded credit for the puzzle. You can also use your tokens you earn in the games to purchase puzzle packs (there is no other real use for the tokens, so might as well use them to buy puzzles). This will allow you to host your own New Crosswords for others to join and complete for you. Sometimes there will not be any New Crosswords in the Public menu, so create your own (making sure to make them Public when asked). It will probably take a couple weeks at this pace to reach 300 unique puzzles to get a Platinum medal, but this definitely requires very little actual effort. Just check back to the game as frequently as you can to earn your level ups and queue up New Crosswords for completion.
This method relies on the fact that there are a significant amount of others playing the game, which has been the case even months after the game's release. I would expect this trend to continue. However, if there some day reaches a point where there is no online community, it would be quite tedious to complete 300 puzzles by yourself. Keep that in mind before going for this completion.
NOTE: Occasionally the game will appear to reset your rank (especially when you are close to transitioning to the next medal). Do not worry, this is just a glitch. Continue leveling up as normal and your level will be correctly displayed in no time.