The DHCP settings shouldn't contain a gateway address in /etc/network/interfaces. To configure DHCP on eth0, for example, you only need the following lines at a bare minimum:
auto eth0
iface eth0 inet dhcp
Once that's configured properly ifup & ifdown should work correctly, then you can put any scripts you want ran when changing an interface's state in /etc/network/if-up.d & /etc/network/if-down.d. However, if your DHCP server is setup correctly it should provide a DNS configuration to it's clients, so configuring it by hand shouldn't be necessary