Your question doesn't sound like strictly actor related, since it's more CQRS vs. CRUD characteristic. In standard CQRS scenario you want to have some kind of persistent storage with schema adjusted to your read model. Then you can run your queries on it.
One of the things that comes to my mind using Akka.NET in CQRS-based but not yet event-sourcing scenario, is to utilize Akka event bus. When you start an application, create actors responsible for updating your read model and immediately subscribe them to Akka event bus for listening messages denoting, that read model must be updated. Then in any actor which may provoke read-side changes, publish a related event/message.
If you're using event sourcing, you would probably like to take a look at Akka.Peristence
. There you have a concept of event journal created by so called persistent actors. It describes the stream of all operations made by an actor, which should have been stored. Second thing is another kind of actor called persistent view, which may be associated with particular stream of events and build read model from them. While persistent view is usually correlated with a single persistent actor, you may again use event bus + listeners to create a global read models. One of the advantages of this approach is that all events are stored in persistent storage, making your system more durable and easier to recover in case of failure.