Can you carry on a full adult conversation in sign? If so, then yes, you probably could raise a full sign-speech bilingual child (bimodal bilingual). And yes, it has similar benefits to raising a child who is bilingual in two spoken languages. The research I've seen on bimodal bilingual kids has all focused on hearing children with deaf parents, but there's no reason a hearing parent who is a fluent signer can't raise a bimodal bilingual child as well.
Currently, there's a movement with hearing parents teaching signs to hearing babies ('baby sign'). Most of these parents are not fluent signers and teach only a limited vocabulary of signs, but even so, it seems to have a benefit. These kids show a slight boost to vocabulary, especially in the earliest stages of language development.
Sign languages seem to have a special benefit as well. Babies learning sign (whether from fluent signers or just baby sign) typically sign their first words a little earlier than most babies say their first words. The reason is because they have enough fine motor coordination to make recognizable signs before they have enough oral motor coordination to speak recognizable words. This earlier communication seems to serve an added benefit of reducing the severity of the 'terrible twos', since one impetus for tantrums is trying to communicate and not being understood.
In addition, some studies suggest that the bilingual bonus is greater the more distinct the two languages are - for example, if they belong to separate language families. Obviously, it's hard to get more different than using separate sensory modalities. Fluent signers also tend to have better spatial skills, because ASL uses spatial grammar. (Similarly, people who speak tonal languages are more likely to have perfect pitch.) Of course, this depends on your own mastery of ASL grammar.
As for how to teach it, just sign to your wife in their presence. That's how most kids learn any language.