In the order to become a Custom Officer you need to appear in the Civil service Examination which is conducted by the UPSC. Here I am giving you the details of this entrance examination:

Nationality:
A candidate must be a citizen of:
A subject of Bhutan, or
A citizen of India, or
A subject of Nepal, or
A Tibetan refugee who came over to India before 1st January, 1962 with the intention of permanently settling in India, or
a person of Indian origin who has migrated from Pakistan, Sri Lanka, Burma, East African countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi, Zaire, Ethiopia and Vietnam with the intention of permanently settling in India.

Required qualification:
The applicant must hold a degree of any of Universities incorporated by an Act of the Central or State Legislature in India

Age Limits
21 years

Tentative Schedule
Notification of Examination November/December of previous year
Conduct of Examination May