To request an IP address from DHCP, simply call the DHCP server with the command dhclient followed by the interface you want the address assigned to. Different Linux distributions use different DHCP clients, but Kali is built on Debian, which uses dhclient. Therefore, you can assign a new address like this:
kali >dhclient eth0
The dhclient command sends a DHCPDISCOVER request from the network interface specified
(here, eth0). It then receives an offer (DHCPOFFER) from the DHCP server and confirms the IP assignment to the DHCP server with a dhcp request.