How can my illustrator and I promote each other’s books?

I assume that you want to connect your books on Amazon for greater visibility, so I'll focus my answer on things you can do there. The list below is what I would consider doing first to last in terms of how effective they are probably going to be; I wouldn't necessarily do all of these items for every book.

• The best thing that you can do is to list him as your illustrator on your book, so that any time somebody looks at his books they could potentially click on his name to see what else he has done and find your books that way. If you weren't involved at all with his books, then there's no real way to reciprocate and get your name listed, but I don't think that it needs to be or should be.
• You can get your books listed below each other in the “Customers who bought this book also bought…” section by having people purchase both books. Depending on how popular your books are, it probably would only take a few people buying both to dominate that section of the site, and potentially land your book in the “Frequently Bought Together…” section right above it.
• If you have people that have purchased the kindle version of both books, and they highlight sections of each book, then you could get them listed together in the “Customers Who Highlighted This Item Also Highlighted…” section.
• You could also create a wishlist that includes both books, or a ListMania! list.
• You could leave reviews on each book that mention the other book and link directly to it using the “Insert a Product Link” button that appears just above the review window when you are typing it out.
• You could start forum discussions about the books and mention the other books in them.

I think the biggest thing to bear in mind is to be useful with your cross-promotion and not spammy; don't mention the other book just because, provide some context. In your lists, provide more than just those books if the category you're creating would work with them, even if they aren't your books.  Do as much promotion together outside of Amazon as you can so that you can push each other's audiences to the other person's books, which in the end is going to reward you the greatest because you'll be sending new buyers to the books and Amazon likes to award products that sell.