As described by Boss when creating the league in 2004; Play all the missions as usual except you can only use the slapper to kill guards and Trev/Ouro/Xenia/Baron/Jaws.
You can use a gun for objective-related things such as locks, alarms, cameras, explosives to help complete objectives non-guard related, or things you have to take out in order to move on such as drones at the start of Control.
You MAY NOT use weapons to boost yourself, warp doors, or to kill guards (duh) and must remain unarmed, but guards can help you blow things up as needed.

Use unarmed only, unless required to destroy an object for an objective or level progression.
Do not use weapons for self-boosts or warping doors.
All guards must be eliminated by using unarmed.
No cheats enabled.

Read more on the Slappers Only forum thread.