Is there a way to have a computer register multiple hostnames or subdomains when registering a computer through DHCP?

Your question needs some clarification. What do you mean by 'have a computer register multiple hostnames'? Normally, when using DHCP, a client computer acquires its hostname from the DHCP server. I don't know of any way that a computer can have more than one hostname.
Same deal with what you mean by 'registering a computer through DHCP'. A client host simply acquires IP and other information from a DHCP server host. The only 'registration' performed is the possible entry of a lease in a table maintained by the DHCP server, for the purpose of consistent & persistent use of the IP for a particular host. No other host or service makes use of that lease record.

It is possible to have multiple DNS entries for a single IP. That is not related to DHCP.

--- rod.

It may be my lack of understand as to how a router adds DNS entries. I thought it tapped the DHCP requests.

Basically I am looking for a way to have a computer get multiple DNS entries in the router without manually typing them into the router myself.

Is this possible?

I think there is some system or convention that uses the DHCP config file as a single point of entry for DNS configuration. I wouldn't mind finding some more details on that for my own purposes, and if I do find something, I'll post it back here. However, I still see a possible problem with the use of multiple hostnames in a DHCP config file (assuming ISC DHCP, or compatible), and that seems like the most likely roadblock to me.

--- rod.

Out of curiosity what would those road blocks be?

And then for the sake of it would something like this be possible if I turned one of my boxes into a dns server and did not use the router?

Using bind can entries be entered through the command line or a service call rather than manually changing a configuration file?

Is there a place to hook for the DHCP client after the computer has received its ip?

The DHCP server configuration has entries that are like
I don't think it is permitted to use multiple hostnames or multiple entries. The hostname is basically an alias for 'localhost', and isn't seen outside of the local host. There are scripts that use hostname and the hostname file, and I assume that these expect exactly one name to be supplied. I would anticipate that adding additional names would probably cause these to break.

DNS entries are separate from hostnames. You can have multiple DNS entries pointing to one IP and this is conceptually separate from the local hostname. I do not know the mechanics of starting/stopping/modifying the DNS server.
On all Linux hosts that I've used (using ISC DHCP) there is a system of hooks that launch scripts triggered by various DHCP client events. You can potentially use and modify these to serve your own special purposes. I have done so, and found it not too complex. See the man pages 'dhclient' and 'dhclient-script' for details.

--- rod.