As simple as it comes: just complete the game on any difficulty. The trophy will pop up after the credits roll. Here is the recommended path to take to take advantage of each Robot Master's weakness (full credit to 8BitArtist for the list):

Boss Order (+ Weapon Weaknesses):

The best thing to do is try and beat the 8 Master Robots' stages in about 30 minutes or less and save your game on a separate save file. On Easy Mode, you can also save in between each of the castle's areas. As long as you advance normally, just killing those enemies that get in the way only and using the Robot Masters' weaknesses against them, you should get this with little effort. It'll take between 20-25 minutes to clear Wily's castle, so allow this much time in your run. The clock stops during visits to the shop, as well. I'm an average Mega Man player at best, and my BLUE BOMBER run ended with 51:24 on the clock.

The second hardest trophy in the game. After you beat the game on Normal difficulty, you'll unlock Hard mode. Complete the game on this mode from start to finish and the trophy will unlock. Some tips on getting through this mode if you are having difficulty would be to repeat beaten stages and "stock up" on Energy Tanks and extra lives. You could also buy the 1/2 Damage modifier, but it is an expensive workaround.

This trophy will also unlock when you attain the MR. PERFECT trophy. In my opinion, the easiest level to attempt this on is Pump Man's stage: there are no enemies that shoot projectiles at you that will damage you and there is no mid-level boss. Just enter the boss's room and the trophy will pop before the fight starts. You can also try for this in a level AFTER you've beaten a boss before (with no boss fight).

Again, you can do this for each level in the first part of the game before the castle - you'll have to get a Game Over once to accure enough screws to buy the Hair item first then re-enter the level - but the easiest method is to buy a 1/2 Damage modifier and the Hair item in the Shop on Easy mode PRIOR to Area 4 of the castle. You'll be taking double the damage without your helmet on, so the 1/2 Damage modifier will negate that status ailment. Just beat the 8 regenerated Robot Masters again in this one level and the trophy pops up before the first encounter with the final boss.

Just defeat 1,000 of any enemy type and the trophy will unlock as soon as that last enemy falls. There are a couple of good farming areas, but the one I used for a bit was in Nitro Man's stage: before you climb the first ladder, equip Blade Man's weapon and dart back and forth before two enemies that dash at you rapidly. Blade Man's weapon will kill them in one hit (if you kill the head and body at once) and they are pretty generous in Screw and Weapon Refill Bubble drops. The trophy is cumulative, so you can accomplish this over multiple playthroughs (confirmed by Smirnoff103).

You must complete this in ONE playthough (credit to Smirnoff103). There are a handful of unique enemies in Sheep Man's, Commando Man's and Nitro Man's levels. If a fast flying enemy passes you that you are sure you haven't taken down yet, just run back so the screen scrolls away from where that enemy showed up: ALL enemies in the game will respawn that way except for the "flip-tile" enemies found in Sheep Man's stage. Once you reach Area 5 of the castle, the little drill bit enemies that spawn from the floors and ceilings are the last enemy on the checklist. Destroy one and the trophy unlocks.

Here are a few that I found that might be easily overlooked:

• In Chill Man's stage, near the end, there is a bird that drops "missles" from the very top of the screen ... make sure to kill the bird and the missle. Try Blade Man's weapon to hit them.
• In Nitro Man's stage, there is a robot that speeds toward you and, when you destroy its body, its head rises away from you and shoots at you. Also if you destroy the head first, the body darts away from you VERY fast. Before you climb the first ladder of the stage, you can sprint back and forth between two of these things. Make sure to destroy each part separately and then the whole thing. Use Blade Man's weapon to make it easier.
• Also in Nitro Man's stage, destroy one of the buses that drive toward you with the Chill Spike (confirmed by dante.jay).
• On Commando Man's stage, there are several: one is a crab-like creature that appears when an "eagle robot" drops and detroys a mine in the sand. So that's 3: kill the eagle (and its projectile), the crab (and its projectile) as well as the mine.
• Also on Commando Man's stage, when you jump over a gap, there is a missle-type enemy that shoots from the gap and then sprouts a parachute when it descends. You can shoot the parachute and the missle will drop. Kill the missle itself while its dropping, the parachute separately and then the whole thing somewhere else in the stage (there are many of these in this stage).
• In Sheep Man's stage, there are Mouse Cursor enemies that draw blocks and then the blocks shoot at you. Destroy the Mouse Cursor and then one of the blocks.
• There are also SEVERAL types of "Hard Hat" enemies that are in every Mega Man game: one shoots a "goo" at you to make you slip and slide around, another runs toward you when shot, another has a little helicopter on its head and still another is immobile. Be sure to kill each type.

Just as it says: use the Buster exclusively on the 8 Robot Masters. You can accomplish this on each of their independent levels OR in Area 4 of the castle. If you are trying for this trophy early on, just be sure to bring at least an extra E-tank with you into the fight ... just in case.

The third hardest trophy in the game. You can't use any E, M or W-tanks anywhere in the game. Don't even worry about collecting them or buying them from the Shop. On Easy Mode, you can max out your lives pretty quickly and an extra life only costs 20 Screws from the Shop, so stock up. Use the Robot Master's weaknesses against them ... they will be the major stumbling block for this trophy aside from the final areas of the castle.

Just beat the game without your number of lives reaching zero. On Easy Mode, the average gamer will get this trophy alongside the WHOMP WILY! trophy because Mega Man 10 is EXTREMELY generous with giving extra lives. You'll probably be capped at 9 lives before finishing half of the Robot Masters levels!

This trophy will also unlock when you attain the MR. PERFECT trophy. Don't lose any lives in the game. Save after each level, bring along a healthy supply of E-tanks and maybe even buy a handful of the items that save you from pitfalls and spikes from the Shop. As long as you are saving after each level on Easy Mode, this trophy is easily attained once completing the game. Just be careful with Mega Man ... there's only one of him!

EASILY the hardest trophy in the game. You have to complete this 2-D, 8-Bit action platformer without taking any damage from anywhere in the game: bosses or regular foes. Save after every "perfect" level, don't lose a single life and don't worry about gathering Energy tanks ... you can't use them. But you CAN use Weapon tanks should you run low on weapon power (confirmed by Linglingzors). Best of luck!!