Run this on your machine
XCopy \\192.168.10.23\c$\YourSQL.bak c:\SQLDestinationDirectory
or run this on the server in a job step just after the backup step
XCopy \\192.168.10.23\c$\YourSQL.bak \\192.168.10.54\c$\SQLDestinationDirectory
If you want to restore it on your new server then you will need to execute sql statements against the SQL Server instance on your machine. This could be done by blind timing i.e. job 1 on the server runs at 10:25 AM and finishes by 10:35 AM Job 2 on your PC runs at 10:45 AM.
Or you could use another job step on the Server to execute the restore command against your pc by using osql
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