Easy "Millstone" achievement
Cast Bitter Revelation and hit at least two creatures in the four cards you look at. Do not place those cards in your hand. "Millstone" will be earned after the spell resolves.
Easy "Relic Seeker" and "Thopter Squadron" achievements
Thopter Tokens are considered artifacts. Put together a quick Red/Blue Thopter deck featuring cards such as Pia and Kiran Nalaar, Whirler Rogue, and Ghirapur Gearcrafter (or any other Thopter makers). Those cards along with normal artifacts should be enough.
Easy "The Roil" achievement
This requires Nissa, Vastwood Seer. A good late game deck is required, as Nissa only transforms when you control at least seven lands. When ready, a good defense is required to keep Nissa, Sage Animist on the board at least four turns. Use her first ability to gain loyalty. When you reach at least 7, use Nissa's ultimate ability to untap six lands and they will become 6/6. Note: There is no expiration rule; if Nissa is put in the graveyard at this point, you will retain your 6/6 land-creature. Attack to do 6x6 damage to your enemy.
Easy "Truth Seeker" achievement
This requires Jace, Vryn's Prodigy. After getting at least five cards in your graveyard, Jace will transform to Jace, Telepath Unbound, which has 5 loyalty. You need at least 6, however. Keep Jace on the board and have at least two sorcery or instant in your graveyard. When possible, use his second ability [-3]: "You may cast target instant or sorcery card from your graveyard this turn". If that card would be placed into your graveyard that turn, exile it instead. Do this twice.
Various General Tips
Decks in Magic Duels must contain between 60 and 100 cards, but you should always opt for the minimum as it increases consistency of the deck, as adding too many cards hurts your odds of drawing good cards.
Typical deck has around 24 creatures, 12 enchantments, sorcery or instant cards, and 24 lands.
Don't stack up too many high-cost creatures with a converted mana cost of five or higher, two or three should do the trick.
It's wise to unlock the Magic Origins set first, as the cards in it are very versatile and will work with various decks.
You should aim to save your removal spells for the later stages of the game to remove bigger threats.
As for the majority of freemium games, you should play the campaign as it will give you enough gold to buy packs, and enough experience to battle valiantly with real opponents.
Easy "Millstone" achievement...