How to set browser to connect to internet using specific device?

[efflandt]

If you have 2 different network connections to two different routers, besides each router having to be a different network on their LAN, they would both typically give you a default gateway, and your computer would tend to follow the first default gateway it comes to in the routing table of your computer.

So unless one of the routers has a way to add a network specific route for an IP range you are trying to reach through a specific router in its DHCP settings (unlikely), you would need to manually configure that network specific routing on your computer itself. Otherwise your PC would always tend to use only one of the routers for all internet access.