You have three separate issues here. First, assigning unique IP addresses to Computer 1 and Computer 2 in the LAN (the right side of your diagram). If the router implements a DHCP server, and the switch is just a transparent switch, then the different computers will be served different IP addresses. This is how routers usually work, so you should be fine.
Second, doing port forwarding on traffic coming into the router from the WAN (the left edge of your diagram). Again, most routers implement network address translation (NAT), so the multiple IP addresses on the LAN appear as only one IP address to the WAN. This means that incoming connection requests will generally be refused, as there's no way for the router to know which LAN IP address to route the request to. If the router supports port forwarding, then you can configure it to route connection requests to a specific external port to a specific port on a specific internal IP address. For example, you could have incoming port 80 requests be routed to computer 1's web server, and incoming port 22 requests routed to computer 2's ssh daemon.
Third, what happens when one of the computers is turned off for a few days, and then turned back on? The DHCP server may issue a different IP address for that computer, breaking your port forwarding configuration. To solve this, many routers support "static DHCP", where they always issue a specific IP address to a specific computer (identified by its MAC address).
So: you need DHCP, port forwarding, and static DHCP to do what you seem to want to do.