As a followup, I’ve been able to kill The Butcher with only 9k life using glass cannon and stacking as much Attack Speed as I can.  It’s a little RNG, but if you can get your Templar to tank him long enough (get him stam!) you can kill him very quickly. You need glass cannon for Act 2 onwards, otherwise you won’t be able to kill the rest of Inferno. Unfortunately you have to time Smoke Screen with a few chains with this method, but he dies so fast it shouldn’t be too bad.

Being the casual gamer these days that I am with my job and lots of side work, I haven’t cleared too far into Inferno difficulty in Diablo 3 yet.  However, last night I took on the Butcher as a solo Demon Hunter.  It was annoying and hard, but fun.  I eventually overcame the encounter and was able to kill him quickly and efficiently.  Actually, very efficiently.  I ended up getting the achievement where you kill the Butcher in under 2 minutes!  Considering many people see the enrage timer, I consider that pretty good!

My big problem was the random-number-generator (RNG) events killing me.  As a Demon Hunter you’re about as squishy as a small bug, so 1 hit does you in pretty well.  Note that unless you have 30,000HP, The Butcher’s chain ability will 1 shot your character.

Knowing this, here are some tips that let me beat him.

This was the build I used, focusing on heavy single-target damage while still retaining some mobility.
Obtain 30,000 life.  Try substituting as few stamina-heavy items as possible over normal gear. Damage is important on the fight. Also, resistance to all or physical resistance items help too, as it’ll make it so you don’t need as much health.
Drop sentry and use Marked For Death on the boss at the beginning.  Keep up a Devouring Arrow tick as much as possible and use Elemental Arrow liberally.  Don’t get too greedy, as he’ll grab you if you’re not ready to strafe.
Strafe to the sides, don’t move straight away from him.  His blue grip chain will always get you unless you’re very far away.
Eat the chains damage.  If you have extra discipline you can time a Smoke Screen to avoid all damage from it.  Chains is random, though, so you have to get lucky.  If he hasn’t done one for a while it’s a good bet that the next ability will be one.
The fire on the ground sucks.  When you want to get to a health well it sometimes comes up at the worst time.  Never stand on it.  You can run over it for 2 seconds with Smoke Screen, but don’t tempt fate.
If your damage is high enough you’ll only have to re-mark the boss and re-drop sentry once during the fight.
Get a strong bow with at least 750-790DPS and a socket.  Gem it with a Green gem for 45% (or more) extra crit damage.  You crit a lot.  Do not get a crossbow.  For whatever reason, crossbows fire slower than bows in Diablo 3.  Seriously, even if they say they’re the same speed, they’re not.  Try it yourself with the same gear on.