Based on what we know so far about Z-Moves, Pokémon can still only learn 4 types of moves at a time. The way Z-Moves are shown to work is that they overwrite one of the holder's moves and turn into a Z-Move of the corresponding type. For example, if my Infernape knew Grass Knot and held a Grass Z-Crystal, I could select Grass Knot for the Z-Move and, because it is Grass type, use Bloom Doom.
There is a possibility that this will change a bit when the games are released, but based on current information Z-Moves are not a way to have 5 attacking types.
Source: Official Pokémon trailers featuring Z-Moves.