Read the classics. These are the epics of Hindu deities and the religious way of life. According to Hinduism, each god and goddess is a representation of the supreme god Brahman. For example Shiva, Ganesha, Kali and Krishna are commonly worshiped in Hindi songs. Study the central beliefs of Hinduism and you can appreciate the music much more.
Practice repeating the sound Aum. According to Hinduism, this is the sacred syllable of god. Hindi songs usually begin and end with the sound Aum as a prayer to the gods and goddesses. Take a deep inhalation through your nose and exhale as you say Ahhhh-ummmm. Become one with it and feel the energy that is attached to it. This centering exercise will allow you to become more emotional when you sing Hindi songs.
Learn and recite a mantra. A mantra is a repetition of sacred sounds and syllables that are prayers to god. A majority of Hindi songs consist of at least one or a variety of mantras. Go to the Jyotishvani website (See References) for a list of mantras. Listen to each syllable and recite the line of the mantra slowly. Add emotion to the mantra to make it more powerful.
Listen to recorded Hindi songs. Some singers in particular are Bhagavan Dass and Krishna Dass. Although these two artists are American, their dialect and pronunciation are much easier to understand than a native Indian singer. Choose a line from a song that interests you and repeat it until it sounds like the recording. Graduate to learning an entire song and sing along to the recording.
Participate in a kirtan. This is a devotional gathering where Hindus sing to achieve union with God. Check the events and services calendar at a Hindu temple or go to the Meetup website (See References) for local kirtans and gatherings in your area.