There is no direct connection from Hammersmith to Basildon. However, you can take the Tube to West Ham then take the train to Basildon. Alternatively, you can travel from Hammersmith to Basildon in around 54 min.
From Hammersmith (Dist&Picc Line) Underground Station Tube to West Ham then train to Basildon. 1 h 19 min £11. From Hammersmith taxi to Basildon. 54 min £151. From Hammersmith drive to Basildon. 54 min £7.