How to get a breakdown of CPU usage at each point in time?
You can turn on Enhanced Monitoring and Performance Insights for more detail by modifying the RDS instance in the portal.

RDS > Databases > databasename > Modify

Also, on an RDS instance you can run Glenn Berry's DMV scripts and most of the queries will work. The stuff that breaks generally centers around drives, the errorlog, agent jobs, and the registry. Just download his scripts for the version of SQL you are running and search on CPU. You can start with the one titled - "Get CPU Utilization History for last 256 minutes (in one minute intervals)". 

https://glennsqlperformance.com/2020/02/04/sql-server-diagnostic-information-queries-for-february-2020/

Depending on the version of SQL your RDS instance is running (must be SQL 2016 or newer), you can turn on Query Store as well.