Replication Options with data subsets
Can this be achieved with replication? 


Yes, Transactional replication can be used in your case.


  Can we get subsets of data from a subset of tables from the source, into our new replicated database? 


Yes you can just replicate the tables (articles) that you want along with its subset of data.

e.g. where PersonTypeId = x --> You need to use static row filter as it uses a WHERE clause to select the appropriate data to be published.

selecting specific articles (tables) :





Filtering what data to publish :




  Would it be something like creating a VIEW on the source, and replicating the results of that view, as a table in the replicated database?


No. See above answer to use static row filter when you are publishing the articles.