Upgrading Jira is a big activity and can range from anything to anything. I would recommend you follow the general upgrade guide by each step.
and then follow the version specific upgrade guide. Say for example you are upgrading fro m4.1 to 5.2 then you need to read each of the individual upgrade guides i.e. 4.2, 4.4 , 4.4.1...5.2. and make sure you follow each of the step.
Upgrade guides on atlassian site is pretty exhaustive and covers all the activities you will ever need to perform.
If you face any specific question please feel free to post it up here
And most importan, do it in a test system before doing it on production:
I added installation directory to this since you mentioned you have customized velocity.
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