Success in securing extramural grant support begins with planning ahead. Find out what materials you need to submit, and start working on your application well in advance of the deadline. It takes time, consultation, and careful revision/editing to develop a strong research methodology and a succinct, compelling, readable narrative statement. You should also plan ahead to ask for any required letters of recommendation, to give your referees adequate time to craft strong letters advocating on your behalf. Make sure to give your referees a working draft of your statement, along with information about the grant you are applying to, deadlines, and to whom/where to submit the reference. There are resources on grant writing linked here for tips and inspiration. You should ask your advisor for additional advice.
Sydel Silverman 1991. “How to write a grant proposal”. Current Anthropology 32(4):485-489. http://www.wennergren.org/news/advice-how-write-a-grant-proposal
Leslie C. Aiello 2012. “Grant Season Journal”, four parts. Wenner-Gren Blog. http://blog.wennergren.org/category/how-to-write-a-grant-proposal/
“Two essays on Writing Fellowship Proposals: SSRC, NEH.” In Cynthia Verba, ed. 2013. Scholarly Pursuits: A Guide to Professional Development During the Graduate Years. Harvard University. Appendix B. http://www.gsas.harvard.edu/images/stories/pdfs/scholarly_pursuits_appendix_b.pdf