You can purchase a license for Acrobat DC from within the application by following the link at the bottom of the right pane.
After your Acrobat DC trial expires, the Trial Expired message with the Buy Now link appears at the bottom of the right pane. Click Buy Now and follow the onscreen instructions.

Yes, Acrobat DC lets you view PDF files even if your trial has expired. You can also use the free tools like Comment and Fill & Sign. But to experience and use the full capabilities of Acrobat DC, you need to purchase a license.
Method 1: For an Acrobat DC plan purchased on the same Adobe ID you used to sign in to Acrobat in trial mode or for a subscription.
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The Sign In screen appears. Click the Sign In button. (If you purchased the Acrobat DC plan on an Adobe ID other than the one shown, click the Not your Adobe ID link.)
If you clicked Not your Adobe ID, the following Sign In dialog box appears. Enter the Adobe ID and password you used to purchase your Acrobat DC plan, and click Sign In.
If you purchased an Acrobat DC serial number, you need to download and install a separate Acrobat installer.
Before installing the newly downloaded Acrobat DC, uninstall the trial version installed on your machine.