This is a fairly tricky achievement to get but very doable.
Golden Axe 1 - Difficulty doesn't appear to matter, but either Easy or Normal should be a simple task. Simply go to the Options (through the emulator) and select Custom, and adjust your Lives and Power Meter to 5. The game shouldn't be too challenging now and you will be able to finish the game. Gilius (The dwarf with the axe) is a nice choice for breezing through.
Golden Axe 2 -Again, Gilius is a good choice due to his weapon's reach. If you are inclined for an in-game cheat, there are cheats that can severely reduce the difficulty of this game:
1) At the opening cinematic, hold down the X, A and B buttons (default controls), then press Start, select Options and press Start again. Remove ONLY the X button, make your adjustments, then press Start on the Back option, then press Start again when you highlight the option to start the game. At the Character Select screen, you can release the A and B buttons. At the bottom right, you will see that you have 8 Credits.
2) This is a little tricky to pull off depending on how skillful you are, or how well you can gauge an boss' life, but if you want to try this out, hold the Magic button down as you finish a boss off, and keep it held, even through the stage's end. With the thief minigame that comes up, just hold the magic button down and do not move. At all. Once it ends, let go of the button and when the stage begins you will immediately cast a spell. This spell gives you the added benefit of having 255 "books" to cast with at will. Bear in mind that if you cast more than your character's maximum magic capacity allows, you will crash the game. Don't use this particular glitch if you want to have some kind of challenge as the game becomes laughably easy.
Golden Axe 3 - The tips in Golden Axe Forever will be very useful, and you may as well get it anyway.
The Panther tactic will be the easiest method of clearing the game fast, and with a respectable enough score to enable you to get the true ending.
If you cleared all three of these games you should get this achievement.
(Thanks to those who commented that difficulty does not matter.)