DHCP v4 traffic operates on port 67 (Server) and port 68 (Client). So we can capture the appropriate traffic with the following expression. (v4)
tcpdump -i eth0 udp port 67 or port 68 -vvv
DHCPv6 uses UDP port number 546 for clients and port number 547 for servers.
tcpdump -i eth0 -n -vv '(udp port 546 or port 547)'