How can you construct as many intersections as possible with n lines?

If you choose $n$ lines randomly, they will intersect at $(n^2-n)/2$ different points almost surely.

More systematically: Choose $n$ directions for the $n$ lines such that no two lines are parallel. Then every line will intersect every other line, and you get $(n^2-n)/2$ points of intersection unless there happens to be some point where three or more lines intersect. So you need to choose the positions of the lines such that no three lines meet in the same point -- that is, for example, choose where each line intersects its normal through the origin (the direction of that normal is fixed once you have chosen the line itself, of course).

Place the lines one at a time. For each line you need to place it such that it avoids all of the crossing points between the lines you've already placed. However, at each point in the process there are only finitely many crossing points, which map to finitely many points on the normal-through-the-origin that must be avoided. But since the normal contains an infinity of points, there will always be points on it that you don't have to avoid, so just choose one of those.