The best way to tighten a clothesline is to use a turnbuckle. A turnbuckle is a device which consists of two screws, threaded in opposite directions and attached by a metal frame.
First, anchor one end of the clothesline to whatever is being used as a post for the line. Then attach the other end of the line to one end of the turnbuckle. Finally, attach the free end of the turnbuckle to the opposite post. Usually, turnbuckles can be found with a eye screw on one side, and a hook on the either.
The turnbuckle can then be easily attached to a post by inserting an eye screw into the post and then fastening the hook on the end of the turnbuckle to the eye of the screw. Once the line is placed, simply rotate the metal frame on the turnbuckle which draws the two screws together, thereby tightening the clothesline.
In addition to using a turnbuckle, the composition of the line can also help to establish a taught clothesline. Cotton lines, although relatively cheap, have a tendency to stretch when exposed to moisture which will cause slackness in the line. To avoid this, use a braided wire line with a plastic sheave. The plastic will reduce the possibility of the wire rusting and ruining the clothes on the line.