How do you actually perform manasik japa? What is the correct method?
You do manasa Japa by contemplating the words of the mantra without moving your lips.
Learned men say that this is Svadhyaya if one performs the Japa of the Vedantic passages, Satarudriya (Vaj. Samhita XVI. 1–66), Pranava (Om) etc. It brings about the achievement of Sattva Guna in men.
Kurma Purana II.11.22
There are three types of Svadhyayas viz. Vacika (verbal), Upamsu (inaudible muttering) and Manasa (mental), those who know the meaning of the Vedas say that the latter ones are better than the earlier ones.
Kurma Purana II.11.23
That which involves only the throbbing of lips but the words of which are inaudible to others is designated as Upamsu (inaudible Svadhyaya). It is better (more efficacious) than vocal Japa.
Kurma Purana II.11.25
The contemplation of all the words in the proper sequence of words and syllables without throbbing of the lips, is called Manasa Japa (mental Japa).
Kurma Purana II.11.26