Place blocks (stone) in the off hand and ladders in main hand. Right click should alternate placing stone and ladders while walking forward resulting in moving up the pillar made.

https://minecraft.gamepedia.com/Tutorials/Dual_wielding#Uses
"Ladders can only be placed next to a block, so your off-hand will place a block on the ground first before you place the ladder on it. Holding the forward key and the right-click button will repeatedly place blocks and ladders while simultaneously climbing up."