Required runway length for landing
The minimum runway length required for landing under Part 135 operations is, believe it or not, the same as Part 91: You need to be able to land and stop within the available runway length specified in the AFM. This is what you use when you are getting ready to land.
That being said, since you reference the "60%/80% rules", there is a more stringent requirement that is required as part of your preflight planning and specified in 14 CFR 135.385
60% Rule for Large Turbine Transport Category Airplanes
This rule is specified in 14 CFR 135.385(b), and says (in part):
no person operating a turbine engine powered large transport category airplane may take off that airplane unless its weight on arrival ... would allow a full stop landing at the intended destination airport within 60 percent of the effective length of each runway
So to follow the regulation by the letter, you need to take 60% of the runway length and compare it to your actual landing distance.
We want to land on a 6,000 ft. runway, so our max allowable landing distance is 3,600 ft.:
Determining minimum runway length
Sometimes pilots prefer to ask the question in a different way, and ask "what is the minimum runway distance needed for the current conditions" instead of "what can my maximum landing distance be. To calculate this, we simply re-arrange our formula above like this:
If our actual landing distance is 3,000 ft. then we would need a minimum field length of 5,000 ft.:
The 80% rule specified in 14 CFR 135.385(f) works exactly the same way, but can only be used by an "An eligible on-demand operator" if approved in accordance with their Operations Specifications (A057 and C049) and the procedures listed in their Flight Operations Manual.
You would then replace the 60% with 80% in the above calculations.
14 CFR 135.385(d) says:
(d) Unless, based on a showing of actual operating landing techniques on wet runways, a shorter landing distance (but never less than that required by paragraph (b) of this section) has been approved for a specific type and model airplane and included in the Airplane Flight Manual, no person may take off a turbojet airplane when the appropriate weather reports or forecasts, or any combination of them, indicate that the runways at the destination airport may be wet or slippery at the estimated time of arrival unless the effective runway length at the destination airport is at least 115 percent of the runway length required under paragraph (b) of this section.
So basically, you use the information from your AFM for wet runway landing distance, or if it isn't provided (since it isn't required), you do the same calculation that you did before but include an additional 15%.
Taking the example from the 60% rule and assuming that it is wet, we have a max landing distance of 3,130 ft.:
Minimum Runway Length When Wet
The minimum runway distance with a 3,000 ft. actual landing distance would be 5,750 ft.: