It's pretty straight forward. You will have to re-submit your channel for publishing every time you want to update your channel with new code. This goes for private or public channels. Follow these steps:
- login to your account and go to the developer dashboard
- go to Manage My Channels
- find the channel you want to update and select Preview and Update
- from the dropdown in the breadcrumbs select Package Upload
- fill out the form, and upload your .pkg file, then click Save
- it should take you to the Static Analysis step next; if not, select it from the breadcrumb dropdown
- click Analyze
- wait for the analysis to complete. you should receive an email when it's done, or you can keep checking back. It usually only takes a few minutes.
- select Preview and Publish from the breadcrumb dropdown
- click Submit for Publishing
Since it's a private channel, it should be published with your changes immediately. You will have to do a system update on your Roku device (or wait until the next automatic update) to see the changes.