See How to upgrade Splunk Enterprise in the Installation Manual. Fundamentally, you just install the maintenance release over your existing installation. The documentation I linked to explains about backing up your existing deployment and what the different procedures are for distributed and clustered environments.
FYI, we've posted an upgrade roadmap that suggests backup steps, order of operations for upgrade, and validation steps with links to the latest documentation to help with upgrade planning. Check it out and let us know if you find it helpful. What's the order of operations for upgrading Splunk Enterprise?
1) Read the release notes
2) Back up $SPLUNK_DB
3) Download 6.5.1
4) Stop Splunk
5) Install 6.5.1 using the appropriate method for your platform
6) Start Splunk
This upgrade is pretty trivial and will likely have no problems.