It is important to note that putts hit across any side slope are uphill on the first part of the putt and then downhill on the second part of the putt.
Once you have assessed whether the put is uphill or downhill (to help you determine the pace of the putt and initial starting line) then focus on the area around the hole where the ball will die and gain an understanding of the direction of the true downslope, as this is where the slope will have the greatest influence on your putt.
By building up a picture of the contours you will build up a picture of the line and pace you will need to hit the ball on for it to go in the hole.
Practice reading putts in the same way you practice your mechanics. This will help you learn to assess the effects different slopes and speeds have on your putts. Building up such experience will help you make more decisive and accurate reads out on the course.