How is the airfoil of a fixed wing ultralight constructed?
There are various ways to build a wing like this. Ultimately you need something to shape the rigid form which you then pull fabric over. The cables do not provide the wing shape they provide support for the length of the wing. This goes back to the fairly early days of aircraft design for heavier than air flight,


  In 1909,Frenchman Louis Bleriot produced an aircraft with notable design
  differences. He built a successful mono-wing aircraft. The wings were
  still supported by wires, but a mast extending above the fuselage
  enabled the wings to be supported from above, as well as underneath.


Mast/cable wings fell out of popularity as spar supported wings became viable in the early days of heavier than air flight. 


  As early as 1910, German Hugo Junkers was able to build an aircraft
  with metal truss construction and metal skin due to the availability
  of stronger powerplants to thrust the plane forward and into the sky.
  The use of metal instead of wood for the primary structure eliminated
  the need for external wing braces and wires


You can use wood like this. Or wood with a foam core.


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You can also achieve the same goal with a metal rib assembly.



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If you have the budget carbon fiber is also an option.

There are a few ways to cover the wing frame once its complete but ultimately the fabric must be pulled over the form and, in some capacity, shrunk to make it taught. 

Fabric covered wings have lead to some of the most successful airplanes in history. There is quite a bit of info out there on how to work with fabric in relation to aircraft. The FAA has a nice handbook on it here and more info here.

If you intend on building something I would take a look at this site about building a Breezy. It looks like you can even get a full kit for one here.