Q:

Bench 225 in two months? 

My current bench press 1-RM is close to 170 lbs.  Is it possible to bring my 1-RM to around 225 lbs in just six weeks (without pills, steroids, etc.)?  If not, how much could I increase my max by at most? 

A:

My current bench press 1-RM is close to 170 lbs.  Is it possible to
  bring my 1-RM to around 225 lbs in just six weeks? 

As with everything, the answer is somewhere between "yes" and "no", or, as I like to constantly put it (stealing from Dan John), it depends. 
If you're a 110lbs female who's been benching for years, then I'd say no.  If you're a 220lbs guy who's only started benching a year ago, then maybe.  How're your shoulders?  If you've got niggling shoulder injuries when you bench, but you've been ignoring them, no.  If they're completely healthy and you already do a lot of back work, then maybe. 
The below is how I'd approach this, on the assumption that a) 225lbs isn't a stupidly massive weight considering your body (as above), b) your shoulders are healthy, c) your bench form is good, d) you can accept that your other lifts are going to take a bit of a back seat. 
First, bench at least 4 times a week.  This doesn't mean you're going to be going balls to the wall every day until your arms shake, what you're going to do is use a weight you're confident with (read as: 6RM), and do sets of 3 or so reps, resting fully between sets.  Try doing 5 or 6 sets and see how it feels day to day. 
The idea behind this is you're essentially practicing the movement, getting your body more efficient at it.  The movement should be smooth and crisp.  It probably won't feel like you're training very hard, if you're not ok with that and would rather leave the gym a quivering wreck (there's honestly nothing wrong with that), then this isn't for you. 
Second, train your back.  Your back should be able to handle a fair amount of volume, and a strong back is needed for a strong bench press.  I like sets of 8 - 12 reps for back work, and I mainly stick to horizontal pulling exercises over vertical.  I also like face pulls and rear delt flyes, as well as bat wings to help strengthen the rhomboids. 
Third, look after your shoulders.  If you get any shoulder pain, stop.  I can't bench with a standard bar anymore because of an accumulation of rock climbing accidents, gym stupidity and a couple of "hold my beer and watch this" moments.  I'm ok with this as I never really liked benching anyway, it doesn't mean I'm ok with anyone else following in my footsteps. 
Lastly, don't completely neglect other lifts.  Feel free to squat every day that you bench.  Like with the benching, you're not going to be doing 20 rep squats every day, do it to practice the movement.  Deadlift a couple of times a week, likewise, don't kill yourself, view it as practice rather than a test of mental and physical fortitude. 
Yes, you can increase the weight over time, but reps should always feel smooth and crisp.  After 4 weeks or so, rest a day or two then test your 1RM to see if it's improved. 
Over time, I've used methods very similar to the above, training every day (or close to it) to dramatically improve my squat (tall guys hate squats), my deadlift (you can still pull with a shoulder injury) and my overhead press (needed to buy new work shirts after this one).  I haven't tried it with bench pressing, but I've read about powerlifters having great success with this kind of approach. 
If you feel so inclined, pick up Pavel Tsatsouline's Power To The People (he does have a specific powerlifting one, which I think is called Power To The People Professional, which might also be of interest). 
All of the above does assume a certain training maturity, if you're not confident with the basic power lifts in the gym, then I'd just stick with a basic program for a while and enjoy the process.