Now that you have a suitable bicycle you will need to build a secure platform for your birds fun cage. I suggest that you mount a platform on the front of your bike rather that behind your seat. First there would be a lot more vibration over the rear wheel than the front with a suspension fork. Second, you will want to interact with your bird and make sure you he is enjoying himself. Also, I think that birds like to see what is coming up, and not be surprised and they can get much more of the flying feel up front and higher.
You will end up with a bit of extra shelving and aluminum stock if you buy from the list below but the next smaller size will not be enough. I think that I spent about $50 or so on all of the items.
List of materials:
1 6 foot section of 16 inch pantry shelving
1 6 foot section of 12 inch pantry shelving
2 3 foot lengths of 1 inch by 1/8 aluminum stock
2 2 foot sections of 1 inch by 1/8 inch aluminum stock
1 24 inch piece of 12 inch by 1 inch pine board without knots, you may need to buy 4 feet at the store
2 U clamps of a diameter to go over your handlebars
4 2 inch stainless steel 1/4 inch bolts and nuts with captive lock washers
1 3 inch stainless steel 1/4 inch bolt and nut with captive lock washer
8 Stainless Steel Washers with 1/4 inch to 3/8 inch center holes
Approx 2-3 feet of solid 12 gauge copper wire to wire together cage
Electric drill with sharp 1/4 inch bit
Hack saw with sharp fine toothed blade
Sharpie pen and a pencil
Wrench for nuts
Wrench or Screw Driver for bolts
Saw horse or other sturdy work surface to cut and drill on.
Large channel lock or vise grip pliers
I was able to find all the items I needed at Manards. You should be able to find what you need there, or another larger home improvement store like Lowes or Home Depot. Here are links to some of the items I used.
U Bolts (Note, this is an example, size will depend on the diameter of your handlebars at the location you will mount the upper supports.)
Coming Next, illustrated assembly instructions.