These achievements are all related to earning all of the gold medals in the game. A gold medal is earned by completing each level rapidly. There is no visible thresholds to shoot for. The only thing that can be said is if you are not getting gold, go faster! Most of the levels are very easy and will only require one try to get the gold medal. Luckily, each level is less than one minute long, so even if you have to retry multiple times, it is not at all time consuming. There is an achievement for unlocking your first gold medal, four achievements for earning all the gold medals on each of the four worlds, and one achievement that unlocks when you earn your 80th and final gold medal in time trial mode.
NOTE: All gold medals must be earned in a single game mode (either time trial or star collecting). Time trial is the easiest so focus on that mode. Thanks to Buckswana for confirming this.