Use Me keyword instead of This. It points to current class instance:

Public Class WebService
    Inherits System.Web.Services.WebService

    Private cUrl As String = "http://www.levelextreme.com"
    Private cUser As String = ""

    <WebMethod()> _
    Public Function MyMethod() As String

        Private cUrl As String = "http://www.anotherurl.com"
        Return Me.cUrl & cUrl
    End Function

End Class

(it is best practice to use & instead of + to concatenate strings).