Dogra is the daughter of Kashmiri immigrants. Her maternal uncle is Dogrifolk music singer Prakash Sharma. She grew up in Baltimore, Maryland and attended Oakleigh Elementary School, Montessori, Dulaney High School, and NYU. She attended NYU from 2000–2004, graduating with a Bachelor's degree in Music. She has stated in an interview that her source of inspiration for music was noted vocalist, Sreetama Ghosh.
In 2005, she formed[unreliable source?] the electro-rock group "Shaa'ir+Func", along with guitarist Randolph Correia. In 2007, they released their first album New Day: The Love Album, followed by Light Tribe in 2008 and Mantis in 2010. She also sang a song "Dooriyan Bhi Hai Zaroori" with Vishal Dadlani in the film Break Ke Baad.
She has made a guest appearance in Rock On!!. In 2008, she was approached by Kiran Rao for the lead role in Rao's directorial debut film Dhobi Ghat - a role Dogra initially turned down, but eventually accepted. In December 2011, she served as a judge for Rolling Stone's Never Hide Sounds musical talent contest. She is the host of The Dewarists, a musical collaborative show on STAR World India.