DAI has very similar system to ME3. You buys chests like you buy packs in ME3. DAI has several additional mechanics like crafting and loot on top of it and you need to craft character armor to unlock that character. Not sure how much crafting plays a role with weapons and other stuff. I posted a link to multi mechanics in the first page.
ME3 multi packs initially contained cards you had already maxed out on. Later patches fixed that and introduced specialized packs for characters, weapons and consumables. Getting all common, uncommon and rare stuff was really easy. The trick was to buy cheapest packs until you had maxed out that tier, then move to next one. This also allowed you to relatively quickly move on to next difficulty and increase rewards.
The people who went for spectre and premium spectre packs without having maxed out earlier tiers had harder time, because their gear wasn't really good enough for gold difficulty and their progression was slower when money was spent on more expensive packs. Again, don't get greedy and aim for the top gear right away.