Enumerate the rationals in [0,1] as q1,q2,... . Plot the points (q1,1) and (q2,1) to start with, followed by joining them by a straight line. Assume inductively that you have plotted the points upto some qn.Then, q(n+1) may lie at the extreme left or right of all the previously plotted qs, or lie strictly between some qk and ql (k,l<=n).In the 1st 2 cases, just join q(n+1) with the previous extreme point, after plotting q(n+1) arbitrarily(keeping some bound in mind). In the 3rd case plot(q(n+1),m), where m is any real within the bound, with m>max(y-coordinates of qk and ql). Then, rub the line joining qk,ql and draw 2 new lines joining qk,q(n+1) and ql. This is just a construction and you can obviously make it rigorous.