A picket fence is an aesthetically pleasing fence typically used for defining property lines. This is a short fence, so it won't provide much in the way of security or privacy. There is just something about the white picket fence that says, "I'm home." Installing a picket fence on a hill isn't much different from installing one on level ground.
Saturate the bottom one-third of your posts with wood preserver and let sit overnight.
Figure out where the fence should be located. Mark off the area with stakes starting at the corners and then space them such that each fence section will fit between them.
Dig the fence post holes with the post hole digger deep enough that one-third of the post will be underground. Put two to three shovels of gravel at the bottom of the hole. Insert the post and then pack with concrete or dirt. Use a level to ensure the post is straight and then brace the post to keep it in place. Pile a small mound of dirt around the base of the post to prevent puddling. Let these set for four to five days.
If using pre-made fence sections, attach each fence section using nails or screws and possibly brackets depending on how the section was made. Step each section up the hill. If the grade is very steep, this is not an option as the gaps under your fence will be too large.
If using individual picket fence boards, attach the top and bottom rails of the fence such that they follow the grade of the hill and the space between them is the same at each end. Nail or screw each fence board to your rails so they all have the same amount of board showing over the top and bottom of the rail.
Make sure you know where your property line is before you start installing a new fence. Call your local government to make sure there aren't any required permits and to learn the local codes if there are any. If the tops of the posts are slanted, rounded, or capped, then water won't collect and they will last longer. The post will be more secure if you make the post hole wider at the bottom than at the top. The bottom rail should be at least two inches off the ground to help prevent moisture issues and makes trimming the grass easier. When using individual fence boards, it will be helpful to run a string between fence posts as a guide for placement.