I am assuming you are using AWS free tier. If so, RDS is part of the free tier and you are allowed 750 hours of Amazon RDS Single-AZ db.t2.micro Instance usage. It also includes 20 GB of General Purpose (SSD) DB Storage and 20 GB of backup storage for your automated database backups and any user-initiated DB Snapshots. Backup storage is free up to 20 GB or the size of your DB storage, whichever is less, and requires you to have at least one DB instance running in your account. If you exceed these limits, or if you terminate all DB instances in your account, backup storage will be charged at standard RDS rates.
The RDS billing details is here. Since you did not mentioned which RDS engine, i am not sure what would be the cost for particular instance as it depends upon many factor like region, instance class, engine, MAZ/SAZ etc which you can find below. As far as your question on "billing while RDS Instance is backing up? The answer is yes. As RDS will be accessible for your usage to make any DDL/DML operation. You can not modify the RDS when it is in backing-up state but you can pretty much use it for your read/write workload as normal.
For Single-AZ Deployments, Multi-AZ Deployment (one standby instance), and Multi-AZ Deployment (two readable standby instances), pricing is per DB instance-hour consumed, from the time a DB instance is launched until it is stopped or deleted. Partial DB instance-hours are billed in one-second increments with a 10 minute minimum charge following a billable status change such as creating, starting, or modifying the DB instance class.
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