Destroy an object in C#

Do nothing. Your reference (obj) will go out of scope. Then the Garbage Collector will come along and destroy your object.

If there are (unmanaged) resources that need to be destroyed immediately, then implement the IDisposable interface and call Dispose in the finalize block. Or better, use the using statement.

EDIT

As suggested in the comments, when your ClassName implements IDisposable, you could either do:

ClassName obj = null;
try{
   obj = new ClassName();
   //do stuff
}
finally{
   if (obj != null) { obj.Dispose(); }
}

Or, with a using statement:

using (var obj = new ClassName())
{
     // do stuff
}