Leeks are notoriously hard to clean; dirt gets trapped in the layers as the plant grows. If you don't remove all the dirt, it ends up in your soup or whatever dish you're making. I cannot remember where I learned this method for cleaning them, but it seems to work.
Remove the roots and green tops where they start to leaf out. You can use the lighter green parts. Slice, chop and/or dice until the leeks are the size you want for your recipe. Place the leeks in a large bowl, and fill with cold water. Stir them around a bit, then let the leeks sit. All the dirt will go to the bottom of the bowl, while the leeks themselves will float to the top.
Carefully remove the leeks with a slotted spoon, don't scoop too low in the bowl so you don't disturb the dirt. Place the leeks on a paper towel to drain, and pat dry with another paper towel on top. Now they are clean and ready to use!