The Dalai Lama is essentially the head of state of the Tibetan government in exile, known as the Central Tibetan Administration (CTA). It’s technically a continuation of the government that existed in Tibet prior to the Chinese incorporation of Tibet.
When China squashed the Tibetan national uprising in 1959, about 80,000 Tibetans fled into neighboring India, Bhutan, and Nepal. The Dalai Lama fled to northern India, where he set up the exile administration, first in Mussoorie and then in Dharamsala, where it remains today. Tibetans in exile and in Tibet view the CTA as their legitimate government representation.