The easiest level to get this in is the first boss fight. You get one time core for beating the boss, and the second time core is for getting the gold medal.
Getting the gold medal requires a time of -1:00, that's negative one minute. There is a total of 123 time atoms to be found in the open, with a few in the boxes. All boxes can be broken by the crowbars that are found at the upper left and right corners of the level.
To fight the boss, you have to get him to hit the 6 electric boxes that are on each side of the arena. Lure the boss to take a swing at you and move out of the way to make it smash the walls. Once you've gotten him 3 times, 5 platforms appear at the top of the arena to cross with robots. get to the right side and make him do the smashing again, this time his arm is faster.
Once you get both arms off, his 2 hover devices expose and 5 more platforms appear under him with more robots, all of these have spikes on their heads. Smash the 2 gold parts under his robot until they fall off, and you'll beat him. This will pop Die Tenorman Die, the t-shirt, and then you grab what atoms are left and hit the exit.
Edit: Due to a request, I will do my best to give a general idea on speed.
From the start you want the left arm to smash platforms where the ladders are. This way you can jump upwards and fall downwards on the ladders. Work from the bottom up. You may want to get your hands on a crowbar from above to smash some of the crates that are under the electric boxes, but be careful not to touch the boxes, as sometimes I would only be standing under the boxes and not jump but swing my crowbar and die to the box without touching it. To be reasonably fast, you need to not waste the punches on walls that don't need smashing. If a box is going to be too far out of the way, you might want to decide if you want that box badly enough or not. Keep in mind that his missiles will also break walls and smash crates.
When you've busted his left arm but before the cutscene, stay on the ladder. If you try to jump off to reach the top, it will letterbox and you'll fall to the bottom. After the cutscene, grab a crowbar in a hurry if you didn't get one, or grab it to refresh yours if you did and smash the easy to get crates, like the ones at the 1-up and the 6 on the platform. If one falls, forget it, it might land on the electric field below the boss and break. You've lost that one, don't worry about it. Sometimes your crowbar will break before fully smashing the 6th box, make a judgement call to get another crowbar. It will help you get across the 5 platforms up top unless you can do precision jumping on the robot heads. These don't have spikes on their heads.
On the right side, grab a fresh crowbar and keep in mind the boss is faster now. Work top to bottom this side and grab what you can. Have him smash the platforms at the ladders again so you have easy jumping and climbing access. Try not to waste punches as you work.
At the next cutscene, you may want to be at the middle of the ladders before it starts. If not, you will need to head for the middle to get on the platforms below the bot. You might also want to grab a fresh crowbar before you jump to the platforms. The little robots now have spikes on their heads, so you must hit them with the crowbar. Each side of his hover device will take 3 hits to break.
After the last cutscene, you are now on the bottom right side, and you can now grab the bottom atoms to top off your count and decide if it is worth going further left to grab all of the particles or just jump for the exit. I jump for the exit at about -1:06, that's negative one minute six.
The one basic tip of try not to die is a huge help here, as you lose a lot of time with the respawn. Practice practice practice is also important, and you can also try to bring a second player. If you do this, you have to be careful that the arm doesn't track the wrong player. If you are both working, you have to pay attention to the arm, because if it hits a player, you won't smash a wall. If you have a player who isn't that great, have them stay at the bottom left or right where the arm won't reach you.
After a few attempts, you'll learn everything there is to know about the boss and you'll be able to run him first try collecting most everything. This isn't something you'll get first try. You just have to get a feel for it. I can't really give you a time thing for it, because you may or may not actually go whole hog for the particles. There are 123 particles available, so that's 2 minutes and 3 seconds to collect, plus the boxes. You can also go after what you missed after the fight and jump for the portal at the end in the middle from above. Just try not to take too long. If you feel the fight is taking too long, restart. Do your best to avoid the arms, make them do the work, and move confidently.
Edit 2: McDurbin went through the trouble to record a video for this solution as well, and I have included it if you prefer to see this done in action. Awesome work!