Use a counting loop with a counter from 1 to 1000. The loop body would use a single Random object to pick random X and Y locations for the center of the circles.
Here is a start on this idea:
To make sure that each circle completely fits within the applet drawing area, make sure that each center's x and y is at least a distance of 5 (the radius) away from an edge. But let us not do that.
Complete the applet by filling in the blanks. The drawing area is going to be 400 by 400.