First, you want to set up the game right. Make sure the number of target lines is set to 8. Set the fill start to 5 so that only one square is missing to clear the first target line.
The target lines are directly above the thicker blue lines on the grid. Once you clear a target line, quickly fill in the open space up to the next target line, clear that one and repeat. Be careful to not clear any lines below the target line or the grid will drop and you will lose any progress you made on the next target line.
The video below shows one of my runs at this: Need to get a little lucky at the end to get the right pieces to clear the last line since you don't have much room to work with. I got pretty fortunate in the video.