For me this is the hardest achievement and I needed several attemps to achieve it. Here some tips that should ease your attempts:
1- Be patient: No time in this mode, so some levels that in Tour mode were a nightmare here are a piece of cake. Besides you have to complete all the levels, including the bosses at one sitting. In my case it took 1h 15 minutes, so be patient.
2- Avoid the food and try to kill yourself often: You start with 3 lives but there are good news. You will earn an extra life (in case that you lost previously some) when you reach some of these scores:
So you have 23 opportunities to restore a live in case you need it. Here is my tip: Try not to earn points (avoiding the food, e.g.) even kill yourself if you can just before to reach one of the aforesaid scores. When you reach any of the aforesaid scores with all the lives you will earn 50K extra points. Of course, hardest levels are close to the end and if you get all the foods and earn always these 50K (easily in the starting levels) when you really need to restore lives you won't be able to due to your score will be above 12.5M.
3- Prepare the second controller: This is the most important tip. Start with the first controller, do your best and when you were with 0 lives (and with no possibility to recover anymore by score), switch on the second controller and start to play with the second player. Good news are that you start from that point with 3 lives again and with the possibility to recover lives again with the aforesaid scores!
Achievement will pop if you beat the mode with the second player, don't worry.
I think that these tips will help you a lot.