Can I just point out the the original post said _Vertical climb_.
In which case any horizontal component of lift has to balance any horizontal component of drag or thrust (or the plane will move horizontally as well). If we are talking about a well designed plane where all the drag and thrust lines act through the CoG then the horizontal component of lift really will be zero when climbing verticaly as he said.
The answer is simply that for 'perfect' aircraft to make a vertical climb the wing has to be flying at an AOA at which lift is zero and the four forces program shows this nicely. If you put the plane into a vertical climb or dive all the horizontal vectors disappear and thrust and drag point straight up and down.
However in most real world aircraft the zero lift AoA is not zero degrees so to make a vertical climb the wing has to fly at a slightly negative AoA.