Who first determined the distance between the earth and sun and what method did he use?

The method used was to carefully time the very rare transit of Venus across the face of the sun. It was 1st attempted by Jeremiah Horrocks from England in 1639 (published in 1661) but he was off by about 100%.

The 1st accurate measurements followed the 1761 and 1769 transits and were done by the French astronomer Jérôme Lalande.

For more details see http://en.wikipedia.org/wiki/Transit_of_Venus#Modern_observations