Traveling with chicken
If you cook it immediately on arriving, an hour is a safe period.  Remember the chicken will take a little time to heat up. Usually 2 hours is the limit for meat at room temperature.

On the other hand, there's no harm in playing it safe, with both spillage and spoilage.  I'd suggest you put the brine and chicken in a zip-seal bag, then put ANOTHER bag around that to protect against leaks, then throw the whole thing in a larger tupperware container with ice packed around it. 

This way you're not going to worry about leaks, and if you get stuck in traffic for longer than expected, the chicken will still be fine to cook and eat.