When you tell SQL Server to run a backup, the backup process runs under the context that SQL Server is configured in, not the context in which you are connecting to it. That context does not have access to the C: drive. It doesn't matter whether you do or not. You need to identify which user SQL Server is configured under and change security settings there.
"The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood"
- Theodore Roosevelt
Author of:
SQL Server Execution Plans
SQL Server Query Performance Tuning