1. A system for adaptive resolution of domain name system (DNS) requests obtained from devices of a virtual private cloud network environment (VPC), wherein the VPC includes one or more virtual computing devices arranged within a virtualized local area network, the virtualized local area network generated by a substrate network hosting the VPC, the system comprising: a non-transitory data store including data identifying one or more rules designated by an administrator of the VPC for handling requests, the requests obtained from devices of the VPC, to resolve domain names into corresponding network addresses, wherein the one or more rules are separate from other rules associated with other VPCs; at least one computing device configured with computer-executable instructions that, when executed, cause the at least one computing device to: obtain a request from a virtual computing device of the VPC to resolve a domain name into a corresponding network address; determine, from a VPC identifier of the request, that the request originates within the VPC; apply the one or more rules to the request to determine a resolution server, designated within the one or more rules, to which the request should be forwarded, wherein the resolution server designated within the one or more rules is a private DNS server implemented within the VPC; forward the request to the resolution server designated within the one or more rules; obtain the corresponding network address in response to the request; and return the corresponding network address to the virtual computing device in fulfillment of the request.