Red is a combination of Agouti dominant A, could be dominant B black or recessive bb chocolate (many breeders find that chocolate makes a better overall red color, as it masks the smut better), full color dominant C, dense color dominant D, and recessive non-extension ee that removes the dark color from the main body hairshaft, leaving the red tones behind.
Non-extension ee also usually removes the dark undercolor, and many red rabbits also have wideband recessive ww, which doubles the width of the red color band on the hairshaft, with the white undercolor. However, wideband on a chinchilla rabbit often leaves it much paler than normal, not good for a show rabbit.
Mating red to chinchilla you will expect a lot of agouti rabbits, as both chin and red are agouti based colors, chestnut agouti a likely outcome. The chins should be black B_, but the red may carry chocolate or be chocolate based, so you could eventually end up with chocolate agouti and chocolate chins in future generations. Red is full color and dominant, chin is the next down in dominance, chd_. Breeding back to chin should increase your chances for chins. Both are dense color (not dilute), so that should mesh well. Non-extension will be an issue, as mentioned, non-extension removes the dark color from the hair shaft, leaving only the yellowish/reddish tones. Chin removes those, leaving a pearly white rabbit with brown or marbled eyes, sometimes with dark tips to the hairs (ermine, aka frostie aka frosty). Not a showable color except for a few breeds.
For good chin color, you want to keep black-based, not chocolate-based breeders. At birth, both chins and chestnut agouti kits are black with white inner ears; chocolate agouti and chocolate chins are chocolate brown with white inner ears as newborns. By ten days of age, you should be able to see the banding on the hairshaft, chins will not have the yellowish middle band on the hair, agouti rabbits will. You want nice dark undercolor on your chins, should be a slate gray. Reds often have white undercolor, so choose kits with good undercolor as breeders.