I wish I could refer you to something but the best I can recommend is google around. I would say from the top of my head that if you make a wooden frame it should be white pine, the wire mesh should be around 1/2"x 1" for the bars and the floor mesh should be 1/2" x 1/2". You might try a feed and tack store. Many of them have parts for building rabbit cages which are pretty much the same construction. The difference is chinchilla cages are taller and have wood shelving in them. A good size would be 2' x 3' x 3'. I'd put shelves in spaced about 6 to 12" inches apart. If you have a kit that is just weaned you may not want it much taller than 17" until it grows to around 400g to prevent injury.
If you are using staples be extremely careful your pets won't chew the wood down to the staples. Chinchillas chew everything they can get a hold of. If you browse our cages board you'll see several examples of home built cages. Mine is a modified ferret nation but a FN-182 with an aftermarket metal pan for chinchillas is what I'd consider an ideal cage for a chinchilla.
Inside the cage you'll need shelving. If you have a wire floor you'll want a metal pan to go underneath. If it's a solid floor pan inside the cage you'll need white pine or aspen shavings in the pan. I recommend an exercise wheel (for a grown kit that is over 450g), no smaller than 14". I recommend a food pan like a cup for a bird that fastens to the bars and is all metal, wood perches and hanging toys, and a gravity fed water bottle.
No plastic should ever be used in a chinchilla's cage. It is dangerous to them.
Be careful of hay feeders. Many types are dangerous to them. If you build one ensure your pet cannot fit its head into any gaps. Wire hay balls are the worst. I just put my hay in one corner on the floor. It's not ideal but it works and is safe.
Of course consider your diet carefully. No pellet or food mixes should be used. Straight alfalfa based pellets such as Oxbow Chinchilla Deluxe should be used. Also you need loose hay such as timothy hay in the cage at all times to keep their teeth trimmed and healthy since tooth problems appear to be the leading cause of death in pet chinchillas diet is critical.
In my cage I also have a marble slab. Granite or marble is something chinchillas like to sleep on to help cool them. (although these are not a substitute for proper room temperature ranges).