Google "bike gun rack" and you'll find all sorts of redneckery. Most opt for, incorrectly in my opinion, a bicycle version of the classic back window pickup gun rack: two basic U shaped cradles mounted on the handle bars. Bike tourists and backpackers know the basic rule to gear hauling: keep the weight mounted low on the bike/pack to keep the center of gravity as closet to the ground as possible. The handle bar mounts not only put a bunch of weigh on the highest point of the bike, but also increase the width of the bike. Try riding a single track trail with that. Not happening.
I find that it's often hard to improve upon tried and true methods. That's why the pictures on the google search that most catch my eye, are the vintage military photos with some WWI era bloke toting an Enfield down the countryside. Fittingly, my rifle literally is of that vintage, 1913. So for my upcoming grouse-bike hunt, I took a hint from history, and mounted it something like this...
Curious how I made mine? Follow along: