There's a pretty easy way to do this, but you need to get past Talrand in the campaign first. There's a lot of guides on how to do this, but it's definitely an annoying feat to do with the rubberbanding AI.
Anyway, the first "Encounter" you run into is a guy who will only use Suntail Hawks, ("Disturbing the Nest"), and this is where you can seize this achievement within 10 minutes or less.
Use Garruk's Pack Instinct deck and make sure your opening hand has at least 2-3 lands and a Brindle Boar or two. These Brindle Boars are huge in the achievement, so make sure you have one or two ready to go. This will also be a lot easier if you have some Garruk's Packleaders in your hand due to the trample effect they have.
Once you eventually mulligan and/or restart the duel to garner a quality hand for this, play the game. What you need to do is get some creatures out and just wait. He'll only summon one hawk a turn, so you'll have a while to accumulate enough mana and creatures to facilitate your plan here.
What you need to do is summon a Brindle Boar and then try to get a Garruk's Packleader out (low cost with trample) and then begin to attack. You can't block this guy's flying creatures until you get out your Sentinel Spiders, but if you don't draw them, just let him attack until you get below 16 health.
Eventually, you'll have a bunch of land and creatures out after a few turns. Just drop whatever you have and amass a huge war drum of creatures there and start attacking. Summon any life gain spells, including the Bountiful Harvest card, which will help you out a lot in this achievement.
If you attack each turn, he will block the strongest-hitting creature and thus kill the Suntail Hawk. It's important that you attack with only creatures above 1/1 so he can suicide his hawk. You don't want him to have more than 3 out there at a time, if he does, it might be a little tough to continue, but probably still feasible.
Anyway, get his health down low enough to where you know he can't block the damage from an inevitable deathblow (Packleader's have trample which will stack the damage past the block, so 2 will go to him). On the turn right before you take your killing swing, use a Brindle Boar or two's ability and sacrifice them to gain 4 life from them. He'll try to block, but the trample effects and sheer outnumbering will be his end.
This is a really easy way to get this achievement and has been done a few times just to showcase its validity. Do it on the Disturbing the Nest encounter and spare yourself a life of trouble and unpredictable AI patterns, this one guy does nothing but cast one hawk a turn and attacks with the ones without summoning sickness. Use his predictable AI to your advantage.
Note: ShadowManGMR's achievement solution for "Savior the Moment" (win by draining opponent's library) could be done at the same time as this game. It will make it take a bit longer, but once you get into a routine, you could easily prolong the game and have him cast his entire arsenal of creatures (with you killing them, due to him using them as blockers) without going below 20 health. Just keep a formula going, one at a time, with every hawk he brings out.