I would recommend attempting this achievement in one of the earlier Zone 1 or Zone 2 levels. The later levels have difficult enemies to kill who will easily knock your rings everywhere. The also have many more sets of spikes. I would recommend taking a look over the Zone 1 and Zone 2 level maps for Palmtree Panic and Collision Chaos using this guide and deciding which level you think you could gather 200 coins without hitting an enemy or a level trap such as spikes. Remember that the past, present, and future all have their own sets of coins and that changing time periods does not rid you of the coins you have already gathered.
Once you have figured out which level you wish to play, you will need to find the past sign on that level, and to travel to the past. Refer to 88 Miles per Hour to see an easy way to travel to the past on any level. Once you have traveled to the past, find the robot generator and destroy it. Once the robot generator is destroyed, all enemies on that stage will no longer be on that stage. After you destroyed the robot generator you should be able to find all 200 coins while staying in the past. Remember, the coin boxes hold 10 coins. Take your time now that all the enemies are gone and continue to use the map provided by the link above to know where all the coins in the stage are. Do not forget that there are 10 minute limits for each stage.