If you want to play older games with achievements there are a couple ways. Either by playing with a modified emulator and fan-created achievements, or games that have been re-released on Steam.
This is the download page for customized emulators made by folks at Retroachievements.org based on GPL-licensed emulators with achievement support included. They will appear as overlays during gameplay. The source code is available here.
Shame however that the achievements proper that can be got from the site aren't that interesting (supposedly due to limitations with the emulator's achievement system), but you can develop ones yourself. The list of achievements are here.
- RAFCEUx - NES emulator, note RANES has changed to RAFCEUx
- RASnes9x - SNES emulator (based on Snes9X)
- RAVisualboyAdvance - Gameboy/GBA emulator (based on VBA-M)
- RAGens - Mega Drive/Genesis emulator (based on Gens)
- RAOotake - PC Engine emulator, note RAPCEngine has changed to RAOotake
- RAMeka - Master System emulator
- RAP64 - N64 emulator (based on Project64)
- RAHandy - Atari Lynx emulator
- RANes: If you have a black screen at startup, change DirectDraw to Hardware Acceleration for Full Screen.
- RASnes9X and RAVisualBoyAdvance: The achievement overlay only currently supports DirectDraw output.
RetroArch integrates with the RetroAchievements service to provide a trophies unlocking mechanism similar to modern web consoles. Keep in mind that not every RetroArch core is compatible with RetroAchievements. Check this list to see wich one to choose. Also check out the this page for easy set-up instructions.
Some older games get re-released on Steam with achievement support, like the Final Fantasy games.
One recent re-release of N20 for the PS1 on Steam was found out to be an emulated PS1 ISO bundled with a modified build of PCSX-R (using HLE BIOS) to support Steam Achievements. Tweaked, it could be the basis for a new build of PCSX-R with achievement support. The source can be found here.