Do I Need To Purchase Premium Boosters to Win at Campaign/Multiplayer

Yes, you can. The beauty of magic is that it allows a lot of flexibility on what you can play of cards, and the synergy between them. In your case the selection of cards will be limited, so your available options for decks are as well.

There are few, if any decks out there that can beat "everybody else's deck", so if you manage to build a deck with either solid synergy, or some other advantage quality (say rush deck, plague deck etc).

You should be able to pull out some decent, competitive decks from the get-go without paying, but sooner or later, some decks will go mainstream, causing your deck to be inferior as "everybody else" sit with decks that can, for example, kill you on second turn.

It is true that there are certain premium cards that are only available in purchased DLC - so, in theory, by not unlocking those decks you are reducing the potential pool of cards from which a deck can be built.

However, The Escapist (via PAX East) reported that there are 'safeguards' in place to prevent players from building high-powered decks purely from the unlocked DLC:

Your collection is limited to four of any common, three of any uncommon, two of each rare, and one of each mythic. This will keep players from just buying a hundred boosters and building the uber-competitive super-decks that you'll see at the tournament level of competitive play. It's a welcome restriction for players like myself, that tend to avoid microtransactions.

I would argue that you stand the best chance of building an optimal deck if you purchase all the DLC - but that's not to say you won't be able to make a competitive deck without them.