Krishna bhakti is to fulfill the desires of Lord Krishna with respect and love. There are two stages, vaidhi, and raganuga. Vaidhi means obligatory, there may not be love or much feeling, but one executes bhakti out of dutifulness and obediance.
Gradually that brings the stage of raganuga, or spontaneous loving devotion which is our natural state of mind.
In Bhagavad-gita Lord Krishna asks four main things. "Always think of Me, worship Me, bow down to Me, and become My devotee, in this way you will come to Me."
In this day and age the specific core practice of Krishna bhakti is to chant or sing the Hare Krishna mantra, and to generally hear and repeat the glories of Lord Sri Krishna. Especially performed before His most merciful Manifestations as Lord Chaitanya and Lord Nityananda.
The complete science of Krishna bhakti can be found in "Bhakti-Rasamrta-Sindhu" by Srila Rupa Goswami.
References:
Bhagavad-gītā As It Is 9.34
Nectar of Devotion (Bhakti-Rasamrta-Sindhu)
Krishna bhakti means to serve Lord Sri Krishna with love and devotion. This Krishna bhakti is described in Bhagavad-gita, (especially the 9th and 12th chapters), and in detail, within the Bhakti Rasamrta Sindhu. i
The cast of Shree Krishna Bhakti - 1955 includes: Lalita Pawar
The cast of Shree Krishna Bhakti - 1973 includes: Jeevan Lalita Kumari
Kathak sources the largest and richest compositions in Krishna's avatar. The reason being that the shringar and bhakti elements in Kathak can be depicted in Krishna's playful life only
Shri Abhay Charan Bhakti Vendanta Swamiji
Mira bai was a rathore princess. from her childhood, she was a great devotee of lord Krishna she popularised bhakti movement by her devotional song which are still popular in India
Dhanurdhara has written: 'Waves of devotion' -- subject(s): Bhakti, International Society for Krishna Consciousness
Hare Krishna is a religious Indian movement who's beliefs are derived from Indian scriptures. It was formed to spread the knowledge of Bhakti yoga which is a practice of cultivating love, submission to God, and complete faith.
You can find several references on web. To start with try this one - " a-milkmaid-called-radha" authored by Devedutt Patnaik
Kenneth Russell Valpey has written: 'Attending Krishna's image' -- subject(s): Bhakti, Chaitanya (Sect), Religious life, Rituals
Yes it is a bonafide movement. It is one of the Hindu religious Bhakti (Worship)movements which spread the message of Lord Krishna what he gave in Gita - one of the most reverred and respected holy book of Hindu Religion explaining the philosophy of Karma (Action or Deed).
Bhakti is the term for devotion to God in Hinduism.