I am not sure what your question is, but here is a quick tutorial on using the width tool:
I would like to convert this contour to a simple line, just an stroke. Is it possible? Then, I would use the width tool.
You could try deleting half of the contour manually and see what you get.
Looks like it has been converted to a fill already which means that you can't use the width tool I'm afraid. You either need to redraw the shape with the pen or pencil tool or else reshape it with the direct selection tool (white arrow) by selection the anchor points and moving them. You might also try using the warp tools, but that would take a light hand, especially if you are not used to them.
Swap the fill for a stroke. Zoom in on each end of the shape, Direct Select the short segment that joins the two sides of the shape (or, if it started as a closed loop, use the Scissors or other tool to cut the path into two parallel paths), ungroup, select one of the resulting two paths, and delete it. Apply the Width Tool to the remaining stroked path. That's basically a method of achieving what Steve Fairbairn suggested.
Thank you all!! Your help was very usefull. However, I think there should be a tool that does it by itself :P.