You'll probably want some DNS queries to be sent via the VPN, in which case you can use split-dns/tunneling and send queries for a configured domain name via the VPN tunnel and the remaining dns queries to the local DNS server configured on the PC.
This FAQ is a good place to start:- https://www.cisco.com/c/en/us/support/docs/security/anyconnect-secure-mobility-client/116016-technote-AnyConnect-00.html
On ASDM go to the Group Policy you are configured to use, navigate to Advanced > Split Tunneling and configure the DNS names, Policy and Network list.
HTH
