See the Wikipedia page on the real projective plane, in particular the section on homogeneous coordinates. The essence of the construction is that a line in R^2 is given by a linear equation ax+by+c=0 (but you can multiply a, b and c by any nonzero constant and the equation will still represent the same line). So we can take the set of all lines with c≠0 and represent each line L by an equation ax+by+1=0. This yields a chart in which L is mapped to the point (a,b) in R^2.
What about charts covering lines of the form ax+by=0? Simple: just shift the origin to a point not on the line and then use the same construction as above.