Clear the game once.
This trophy is best attempted on Easy Mode. If you're really having trouble with any of the levels, just keep getting Game Overs until you have enough Screws to buy helpful items like Energy Tanks and Extra Lives, although using Tanks on your first playthrough will result in more playthroughs later if you are going for the 100%. Beat the eight bosses and defeat Wily twice, and this trophy will unlock after the story is brought to a close.
The recommended order for the bosses and their weaknesses is as followed (although bear in mind that exploiting the bosses weapon weaknesses will result in you not getting TRUSTY SIDEARM):
Sheep Man (Mega Buster)
Pump Man (Thunder Wool)
Solar Man (Water Shield)
Chill Man (Solar Blaze)
Nitro Man (Chill Spike)
Commando Man (Wheel Cutter)
Blade Man (Commando Bomb)
Strike Man (Triple Blade)
Clear the game in under an hour.
This is quite challenging, but perfectly manageable with practice. When you've done a good run of a level, make sure to save your game! A "good time" for a level is on average less than four minutes, but don't worry, there is quite a bit of variation on this - some levels take about 5 minutes, and some only 3, so as long as your average time is on track, don't worry too much. Your time after you've beaten the 8 bosses should not be above 35 minutes. If it is, it would be an uphill struggle to get this trophy, and I recommend you start again. The game gives you ten save files, and I suggest you use them all for this, so you have several points to fall back to if need be.
Clear the game on HARD.
When Mega Man 10 says "Hard", it means "Seemingly Impossible". This is extremely difficult, and what's worse, it needs a Normal playthrough to even unlock the opportunity of playing on Hard. Make sure to save regularly, and the usual advice of getting Game Overs and farming screws until you have enough to face a level applies. Essential items here include Extra Lives, lots of Energy Tanks, the one that makes you take half damage, and the one that saves you from spikes. Lots of them all. If you saved up for a ton of items, used them all and still failed, then reload! Make clever use of the weapons that allow you to jump on your dog, and surf on his back, as they can help you avoid nasty hazards. Persevere and follow this advice, and you too can beat Mega Man 10 on Hard Mode.
Make it to a boss room without getting damaged.
If you're going for the 100% on Mega Man 10, this will come as you attempt MR.PERFECT, but if you're not, then this trophy is difficult but not impossible. Play Pump Man on Easy, as there are no enemies that fire projectile weapons at you - the purple slime shot by the dudes in hard hats doesn't count as getting damaged.Take it slow and steady, and keep out of danger, and this will unlock just as you prepare to face Pump Man.
Defeat the 8 bosses without your helmet on.
Of course, it goes without saying that you can do this by defeating each boss at the end of each stage without your helmet on, but to do this, you'll need to Game Over so you can afford the Book of Hairstyles. The much easier (and recommended) way to do this is to get to the first level marked by a skull in Wily's Castle, and then buy a half damage powerup and the Book of Hairstyles. Using the half damage powerup in this level cancels out the effect of having no helmet on, so you take the normal amount of damage. Defeat all the bosses in this level (which you have to do to advance) and you'll unlock this trophy easily. Then breathe a sigh of relief and switch back to the Helmet.
Defeat 1000 enemies.
This trophy is a blessing, as it can be done entirely on Easy and progress carries across playthroughs - once you've killed a grand total of 1000 enemies across all playthroughs, this will unlock. If you really wanted to farm this, then pick a level with infinitely respawning enemies such as Nitro Man, and kill them all dead until the trophy pops. The recommended thing is to ignore this trophy, and only worry about it if it's the last trophy you don't have.
Defeat one of every type of enemy.
This trophy is, to put it politely, a pain in the behind, as progress doesn't carry across playthroughs, so you have to do it all at once. Fortunately, you can replay stages you've already completed to get any enemies you've missed except the middle levels of Wily's Castle - make damn sure to kill all types of enemies presented to you in there the first time, and save lots as you go through. Do this on Easy Mode to preserve your sanity.
There are a couple of particularly annoying enemies, which I will list here for your viewing convenience:
That awful flying bird thing in Chill Man's level. There's only one of these in the whole game on Easy Mode (though it respawns), which is just after you return to the icy surface after being underground for a while. It flys past and drops a bomb that flies towards you. The easiest way to kill it is to use your dog to jump up to its level as it flies past, then shoot it whilst in the air.
The strange spring-like enemies that freak out and shoot bullets everywhere when fired upon with most weapons - they're not unkillable, just use Chill Spikes.
The trains in Nitro Man's level - again, not unkillablle, just use Chill Spikes.
Apart from that, just kill everything you haven't killed before, and you'll get this by the end of the game.
Defeat the 8 bosses with just your default weapon.
This is best done on Easy, as the bosses become cruelly hard on Normal if you're not exploiting their weaknesses. Check your selected weapon before you go into the boss room, as you don't want to accidentally fire a different weapon, or spend time changing weapons in the boss' room. Just avoid the boss' attacks as best you can, using energy tanks if you have to, and fire only the default weapon at them.
NO COFFEE BREAK
Clear the game without using any energy, mystery, or weapon tanks.
This is quite difficult, as there are situations where Tanks come in incredibly useful, but there are a couple of ways you can make this easier for yourself. Playing on Easy is the most obvious, but using the half damage powerup will mean you can take twice as much damage before you die, often eradicating the need for energy tanks. Enemies often drop orbs that restore life, so try grinding enemies if you need a quick life boost. Try and save your lives for the boss fight, as it's where you'll really need them, and if you're running short of lives, just Game Over and you'll get some free.
Clear the game without continuing.
This makes the game much more difficult, as the old trick of Game Overing until you can afford powerups is out. Playing on Easy is a must. Instead, keep saving, and try not to spend money on powerups unless you really need it. If you Game Over, reload, and you'll be fine. The Extra Life pickups found in quite a few stages are very useful to you now, so at least make an attempt to pick them up without endangering yourself too much - after all, losing a life to gain a life is pointless. Keep persevering, and cautiously saving, and you'll get this without too much trouble.
Clear the game without dying.
Make sure you're playing on Easy, as the amount of instant-death hazards is dramatically reduced. There are two words that sum this trophy up: "Energy Tanks". Lots and lots and lots of them. As many as you can afford. If you're about to die, quickly hit the Start button and use one. Keep saving in between levels, and pick up all the screws you can find to buy more energy tanks. Grinding enemies for extra Life pickups is useful as well.
Clear the game without getting damaged.
I am willing to bet that this is the hardest Silver you will ever get (or not get). Playing as Protoman is highly recommended, as his shield can save you from damage at the last second, and as such is almost essential to this trophy. Saving after each level and playing on Easy are both equally essential, as is exploiting the boss' weaknesses to kill them as quickly as possible - after all, a dead boss can't ruin your attempt at this trophy. Be very very cautious in everything you do, take it as slowly as necessary, and remember - as tempting as it might be to throw your controller/PS3/television out the window/at your dog/into the fiery depths of hell, it is not worth it in the long run. Really. Best of luck. You will most certainly need it.
These videos, by YouTube user Rockman2900
show a Mr. Perfect runthrough of the game.