Hey Terry-- take it easy on Charlie... he's really proud'a them shoes!
Mel- Besides what others have said about going to Ag-treads, etc., I'll toss in my three-cents' worth, as I sure dealt with that issue mowing my dad's hillside as a kid. Depends on how 'steep' the incline is, but here's what I got...
If he's slidin' downhill, the tires aren't biting hard enough. The function of TURF treads is to 'be kind' to the grass, so that it doesn't get opened up for erosion,etc. More aggressive treads, or chains WILL help... and in some cases, it's possible for a more aggressive tread to do LESS damage to turf than a turf tire. I experienced just this situation with my zero-turn- where the turf-tires spin, AG-treads keep good footing, and it rips up less grass as a result.
However, if the grade is so steep that he's slidin' alot, going to more aggressive tires MAY incite a rollover by biting, when turfs would just slide. The solution to that, is running duals. Dual rear wheels act as outriggers, spreading the footprint to prevent rollover. With that much weight-transfer in the lean, he'll likely need steering-brakes to keep the uphill wheel from spinning, and make the darned thing turn. Running chains and half-a-tire full of fluid on the INNER set of tires will do quite a bit for makin' it more goat-like on the inclines... but you're getting into risky territory when you get that close to being-the-horizon.
As Charlie said- goin' straight up and down the grade may be your best bet... and with chains on the tires, it'll claw it's way right up darned-near anything... it may do a little damage on the way, and you may have the urge to get wheelie-bars and a tether-type kill switch, too, but it'll climb. I've got a short incline in a lot up the street here... probably about a 50-55 degree angle, and my 109 will NOT climb it with turfs, but will with AGs, and will pull a trailer up it with chains. Note that with chains, it really doesn't matter if you're running AG tires or not.
It may seem silly to run AG tires on the front (since MOST of these tractors aren't 4wd (ahem... yet), but the tread DOES improve directional stability in the steering axle where the 5-rib 'turf' type tire will slide all over.
Since the Kohler is splash-lube, expect oil-puffing and don't be surprised if you get accellerated wear.