First some basics:
There are IP address ranges designated for public and private use. Private IP address ranges begin with 192.168.x.x, 172.16.x.x or 10.x.x.x. These private network address ranges can be used by anyone for internal use and have no direct representation on the Internet, whereas public IP addresses are assigned to network providers such as ISPs who in turn assign IP addresses to their customers either dynamically on a temporary basis or on a fixed and longer term basis.
IP networks are divided into subnets. The devices within a subnet can talk to each other directly. If they need to talk to a device that is outside their subnet, they need to send the traffic to a gateway that participates in more than one network and relays the traffic back and forth between the networks. There can be more than one gateway in a network (for connecting to more than public or private network).
Routers are a special type of gateway. A router is typically connected to more than two subnets (yes, more than TWO, that's three or more) and maintains a routing table, that is, a list of neighboring nodes, and forwards traffic according to the appropriate neighboring node depending on where the traffic needs to go, the availability of other nodes, and the weight (or cost) assigned to various routes.
Home security routers do not maintain a routing table because they connect only two subnets: The internal network using a private address range and an external one maintained by the ISP which happens to have a public IP address so that traffic can be exchanged with the Internet. For this purpose, home security routers perform an important function called Network Address Translation (NAT) which allows multiple computers on a subnet to appear behind a single, public IP address.
So, now to your problem:
You can give a computer more than one IP address (either on the same or on a different subnet) on the same network interfaceby means of an "alias". In most cases, these IP addresses have to be manually assigned and become static IP addresses for that device (as opposed to IP addresses dynamically assigned via the Dynamic Host Configuration Protocol DHCP). The trouble is, if what you call an "external" address is assigned to a computer, there is no other device acting as gateway and, as a result, this system has nobody to talk to and cannot be reached by other devices.
I'm guessing (because you don't explicitly state so in your question) that what you might be trying to do is getting one device on your private network to respond to requests from the Internet. This can be achieved by means of port forwarding on the home security router. Port forwarding can be selective for specific traffic only, or generic (all incoming traffic from the external subnet that is not return-traffic for NAT from other devices gets forwarded to the device designated to handle such traffic, which otherwise gets blocked by the router).
You should be aware that there is a security risk with port forwarding: The device to which the ports are forwarded to becomes vulnerable to network intrusion attacks. So, make sure to protect that system with appropriate security measures (firewall and up-to-date security patches).