Both methods of chanting—silently in the mind (Manasik Japa) and aloud (Vachik Japa or Kirtan)—are highly effective, but they serve different spiritual purposes. The "best" way depends on your current state of mind and your environment.
When you chant the verses aloud, you create acoustic vibrations. These sound waves positively affect your immediate environment, clearing negative energy from the room. Furthermore, chanting aloud forces your mind to focus on the pronunciation and the rhythm, which is excellent if your mind is highly distracted, anxious, or if you are feeling sleepy.
Mental chanting is considered the most powerful form of Japa in advanced spiritual practices. Because there is no physical sound, your absolute focus is required to form the words in your mind. This leads to deep meditation and an internalized connection with Lord Hanuman. It is best done when you are already calm and in a quiet environment.
A great practice is to start by chanting aloud. Let the sound vibrations clear your environment and focus your wandering thoughts. As you become more engrossed and your mind calms down, lower your volume to a whisper, and finally transition into completely silent mental chanting.