Start with confirming your backups. I've not had many problems out of Splunk upgrades, but that doesn't mean you wont.
Have you read through the upgrade docs?
If you are non-anything (no cluster, all-in-one sort of environment) then you just upgrade using whatever method you originally installed it with (windows is a point-and-click, tar is stop splunk untar then chown the directory again and start, rpm/deb is standard for those tools) .
If you are distributed, there's an order to upgrading the pieces but otherwise the individual installations are as above.
That's all in the docs, so I'd say make a backup, test that backup, read through the documentation provided then give it a try. If you get stuck or have specific questions about the upgrade ask again!
Keep in mind that in addition to the binaries which are being upgraded, the default directories with the explicit configurations, are being upgraded for each component. So, you would like to be in a position where you can compare the pre-upgrade default configurations with the post-upgrade default configurations - fascinating as it's all explicit.
We flipped out a bit during the upgrade to 6.5.1 - Why does the upgrade to 6.5.1 touch SPLUNK_HOME/etc/system/local?
So, it's a good idea to check whether local files get touched and if so in which way and why.