How much money do I need to invest for that?
It used to be that you could buy the entire set for $50 but that is no longer the case. There are 276 cards in the game. I don’t remember where I got this information, but I believe someone said it would take 433 chests in total to gain all the cards. That is the equivalent of 108,250 gems (5 21,000 gem purchases and 1 3,250 gem purchase) or a little over $260. It will almost assuredly be less than that considering that you gain a large amount of packs in the beginning of the game. This combined with the Memoria you would gain by disenchanting duplicates would bring this down some more. Again this is from I number I vaguely remember seeing so take this with a grain of salt.
As a bonus to gain a full mythic collection using just crafting it would be
There are this many cards at each rarity
Which is 162,850 Memoria in total.