You are planning for an MA. The basic eligibility would be having a UGC recognized bachleors degree or above from any university.
IGNOU is a popular and reputed open university that offers M.A Hindi program. It is a two year program that has a maximum time of 5years to complete. The fees is around 9000/year with no age limits. IGNOU MA Hindi program has teleconferencing support and Radio counseling, wherein students can ask questions via a toll free number from home itself and get questions clarified from a specific FM station.
School of Humanities,
Tagore Bhawan, Block F, Indira Gandhi National Open University,
Maidan Garhi, New Delhi - 110068. India
There are many other universities in India that offer MA Hindi correspondence mode.
Kurukshetra University, Haryana
Punjab University, Chandigarh
Nalanda Open University, Bihar
University of Mumbai
University of Delhi
Madurai Kamaraj University
University of Kerala
Once you select the university of choice, you can contact them via their website and submit the forms. IGNOU has many study centers in all major cities and towns wherein study enrollment forms are accepted and help in the admission process. If you choose a private or a deemed university, please check the UGS website that it is recognized.
Some private centers claim to be affiliated to a prestigious university and dupe student of their money (http://www.thehindu.com/news/cities/bangalore/study-centre-offered-fake-degrees-within-a-month/article8439173.ece). Please be careful before you pay fees or cash.