Hi there, here's a how to to add the chicken wire to the doors: https://salvagedinspirations.com/replace-glass-with-chicken-wire-the-safe-way/
And here's a tutorial all about painting a hutch out with chalk paint, hope these links help! Happy spring!
If you want to use chicken wire on the doors, first remove the doors and cut out the inner panel (Drill into each corner to give you a start) Replace Panel with chicken wire and trim with timber to cover raw edges.
Chalk paint can be painted directly on to furniture without preparation, except to remove metal handles or anything else you don't want to be painted etc. Just seal with either clear or antique wax when dry and you are happy with the paintwork.
Enjoy your project!
Here are some great videos and information.
Prep is the key. Clean it first. Sand it and/or strip so you have a good surface to paint. If the original finish is a wood stain, I always seal it before painting so I don't get bleed through.