It is vital to distinguish between a full localization and a subtitles-only patch. The Call of Duty Advanced Warfare German Language Pack typically includes three tiers:
Consoles do not have a "language pack" as a separate DLC.
Before diving into the technical "how-to," let's address the "why." Most modern games include language options natively. Advanced Warfare is a decade old, and depending on where you bought your license, German audio and subtitles may be locked or missing.