Well, I have an account with +500 packs opened (all gathered in 1 year and half) and I am far to finish any set, so you need more than one year botting to try to complete the collection.
Of course, If you are a good player playing arenas, the number of possible packs to be opened increases, and you could get the enterily collection faster.
If I am not wrong, HS has 50 legendary cards which you can get from packs. If you are lucky, you will get one for every 20 packs, and if you are very lucky, you will get all different. So you will need 1000 packs. Of course, the premise that I've used is not correct, because you can create missing legendaries from dust, but it sshows how dificult is finish HS collection without money.
If you want to finish it, never craft a rare or common card, and of course you must dust golden cards to craft legendaries/epic ones.