First, you will need to get a zoning permit (zoning compliance permit) from the Town of Clayton Planning Department. Planning Department staff will help you determine the location for your fence and make sure you do not encroach into any easements or buffer areas.
Also, if you build a fence on the property line, it becomes shared property between you and your neighbor(s). Please note that unless you have recently had your property surveyed, your estimate of the location of the property line may be off by a few inches.
Not every type of fence material is permitted - talk with staff to confirm that the type of fencing you want to put up, and the desired height and location, is permitted.