Gotham Knights

As others have already stated, you will beat five Man-Bats in the course of completing the story, so there's no need to go for this achievement until the post-game. One crucial detail seems to be missing from the other solutions, though:

Man-Bats are not limited to spawning on the three roofs where they spawn in the story missions. They can spawn on any flat roof big enough to house a fight, all across Gotham. To find the spawn locations, you need to check the map to find the League of Assassins faction icon (a circular shape formed by overlapping curved blades).

Go to the district with the League faction presence, and do an IR scan. This will let you see the white – not yellow, that's only in the story missions – crime triangles overlaid on the environment. Any triangle appearing on top of a building is likely to be a man-bat encounter. The man-bat roof locations will have some corpses (white outlines) lying about, as well as some black fog billowing across the rooftop (not visible from afar).

You might not get all 5 man-bats in one night, so when you've rinsed the districts with League faction presence (the League icon will disappear from the map), just pop back to the Belfry and start another night. This will reset and refresh the faction presences.

Edited to add: I haven't confirmed this myself, but several commenters have stated that you can do some light save-scumming to help speed along your post-game Man-Bat hunt. According to Mr SpinelesS, you engage with the Man-Bat "until it scream summons some assassins, then kill it and return to main menu. (leave assassins alive so you don't finish the event)."
You then continue your game from the main menu, and return back to the spot on the map where you just killed the Man-Bat. A new one should then have spawned in its place. This means you'll only need to find one Assassin district on the map, and one Man-Bat building, thus saving you quite a bit of time.
Big thanks to Mr SpinelesS, as well as rhmdy, Crinkle Crisps and AllysonDuck for posting confirmations!

You beat 5 by completing the story. Don’t start New Game Plus and instead roam the city. Each night you start has a chance to spawn one at each of the roofs of the locations you fought them in the Story: Gotham City General Hospital, Elliot Center, and Waynetech. It took me about 10 nights to get the remaining 5 I needed post story.

You'll have to defeat five Man-Bats as part of the main story, so that only leaves five more for the achievement. Once you've defeated the story related ones, you can roam around Gotham looking for the remaining five. Their spawns aren't limited to the three story related spawns, and in fact might not spawn in those places again. It entirely depends on where the League is active. Just scroll through your map and go to any district that has League of Shadows activity. You'll know the League are active in a district if their icon appears under Faction Presence. Simply go to these districts and search any of the high rooftops. If there are some dead bodies on top, hang around for a few seconds and a Man-Bat will appear.