The only place on a golf course where you are guaranteed a level lie is on the tee box. Everywhere else: the greens, the fairways and the humps and hollows around the greens are a lie lottery. When you find yourself chipping off a downslope onto the green, there are a few adjustments you need to make with your stroke to get it right.
Playing off a downslope, the tendency is for you fall forwards. Counteract this natural tendency by standing a bit taller and putting a bit more weight on your back leg.
Make sure that you follow through down the slope. Imagine running the leading edge of the club down the slope as you follow through. Doing this will help you avoid the tendency to stand up out of the shot and chop off the follow through. If you don't swing down the slope, you will likely thin the ball. Lastly, stay down with the shot and keep your head down until after the ball has gone - this will help you swing down the slope and stay with the shot.
When chipping off a downslope, there is a danger that you catch the slope on your downswing before you hit the ball. To counter this, put the ball back in your stance so that you catch the ball first before the club contacts the ground.
Off a downslope, the ball will tend to come out low so choose a club with enough loft to get the ball in the air. Select either a pitching wedge or a sand wedge. If you use a less lofted club like a 7-iron, you will have trouble getting the ball in the air.
Pick the ball up quickly on the backswing or you will hit the club into the slope when you take the club back. The way to do this is to break the wrists quickly rather than sweeping the club away on a wider arc.
When chipping from a downslope, bear in mind that if you don’t land the ball far enough, the downslope will kick the ball forward. Use this margin of safety to your advantage and err on the side of too short rather than too long.