The answer to this is VERY dependent on your network, and likely also your 2 ISPs.
Generaly speaking to run 2 ISPs "properly" you need BGP Routing, in that case your IP configuration won't change - the same public IP will work. The advantage of this is your web site will stay up.
It sounds like you are describing 2 different IP subnets going to your web server - This technically will work fine, but it means your customers will need to realize www.example.com is down and use www-2.example.com (different public IP). Customers aren't goign to realize this though.
Another option is to do this with multiple A records in DNS, but that's a bad solution too. Yet another option is to run DNS servers on each conncetion, each answering with the public IP for that connection - however that's not pretty.
You really need your public IP to stay the same and TRULY be multihomed. The only economical way to do this is to host somewhere professional that has multiple connections - Short of that it's not going to really do what you want cheaply.
As to HOW: If you're just talking one site on the box just leave IIS bound to the default / all IPs (default setup) - Then on your NAT device forward to the IPs - i would put the server on 2 subnets, i.e. 192.168.1.2 and 192.168.2.2 - then use .1 of each subnet for the routers. This way you can use IPs on either subnet for redundancy for outbound connections if you need to - i.e. some machines go on 192.168.1.* and 192.168.1.1 for default gw using connection #1, others use 192.168.2.* and 192.168.2.1 for default gw.
Sorry this is long, it's NOT a good setup and that's the majority of my answer. There are some devices I believe that do this.
Another option that's better is find a host that can forward this across a vpn to you - Why you'd want to do all this instead of just host the site I don't know but by using a vpn tunnel the host could have 2 paths. I've questioned if there's a viable business model for doing that type of service for a long time but don't believe it is - The cost to sell/support it is too high from the hosts end and the cost savings would be trivial, if any at all, given virtualization reduced the cost for a cheap dedicated server.
Good luck,