In order to upgrade to 5.3.x you should upgrade to the most recent 5.2.x version first.
If you have custom modules (or tweaks done to Custom Skin module), you may have to adapt these modules manually (as the upgrade routine doesn't know how to do it automatically). The mentioned link explains how you can convert custom .tpl templates to the new Twig format used in 5.3.x. But you may have to adapt the PHP source code as well (in case of custom modules).
Before upgrading your store please create a full backup copy including the MySQL database and all files. Then deploy this backup on a local computer and try the upgrade there first. If everything goes well, you may upgrade the "live" store on your server. And if something goes wrong this time, you always can recover your store from the backup.
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Is it possible for me to install stock 5.3.1 then migrate my 5.2.15 database?
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The built-in upgrade routine does the migration. You should not do it manually.