The cost of the trip is relatively cheap as far as trips go.
Of course there is the cost of the gas - couple hundred bucks in my truck. The train runs up and down every "weekday" so we drive as far as Cochrane on the Sunday, overnight in a fleabag motel (75'ish bucks) and then take the train Monday morning (leaves Cochrane at 9am, they start loading gear at 7am). The first 40 pounds of crap (per person) is included in the ticket price, after that you pay by the pound - whether it's two or three of us with camping/hunting gear, it typically costs us an extra 40 bucks in "over weight" (I think it may max out there, never asked and always seem to cost the same) - they don't charge to carry out any game animals and you have a whole boxcar - so room is not an issue).
The return cost per person on the train is just on 100 bucks each (it does vary a bit depending where you want to be "kicked off" - they have 13 "whistle stops" and you pay the rate based on the closest one to whatever mile marker you want off at - they will let you off literally wherever you want. From Cochrane to Moosonee there is a little over 180 miles of track that follows the Abitibi then the Moose river all the way up.
We generally stay up until Friday - get picked up about 5:30 pm which puts us back into Cochrane about 11 pm - so by the time we get unloaded/reloaded we stay overnight in Cochrane again (another 75 bucks) - plus it's nice to get a shower
We used to go up on the Friday (for the Saturday opener which is always the 3rd Sat in September and bear/birds/bunnies is already open) but as we have got older/softer we have found that five nights is "enough"
So basically, for two of us the entire trip (gas, train, hotels, moose/bear/small game licences) comes in under $800 total and that's driving about 1600 kms round trip. We are camping on crown land, so no costs there. Logistically it's a fairly simple trip. If you plan on taking up a canoe they used to charge an extra 100 bucks for that if they have the canoe carrier car hooked up - we have taken a canoe up a number of times - if they don't have the car, then they have just added it as "over weight" and we tossed it right in a boxcar (for about half the cost - they charge extra because it's a pain to get it off the canoe car racks if it's in use - or at least that is their reasoning). On a couple of occasions we have seen guys bring up 12 or 14 foot aluminum boats with motors - if you can fit it in a boxcar, they will carry it for you no problem.
It's always a coin toss as to whether there will be enough water to float a canoe - the Moose river itself is usually fine (but wasn't in 2015 - you could walk across it in ankle boots), but along the way from Cochrane to Moosonee there is close to a dozen rivers that can be paddled if the water levels are decent. This year, despite the dry summer, by the time we were there in September there was decent water levels. But if you were looking to plan a trip a call up to the MNR (they have an office in Cochrane and a field office in Moosonee) can quickly answer your questions about water levels.
The bugs can be bad - particularly the black flies and the weather, which comes off James Bay can be nasty but if I was going up to canoe I would probably go in late August - weather is settled and the bugs are mostly gone. I have been up there in June and July and it was just "nasty"
Although you are fairly isolated you are never "alone" - there is enough activity on the tracks between the train and then the crews that inspect the entire line twice a day (pick-ups that are rigged up to drive on the rails) that if there was some kind of emergency you wouldn't be waiting days (unless it was over a weekend) to get some help and the crews "do stop" daily if they are aware you are camping etc to make sure everything is ok (Ontario Northland keeps track of who is where as much as they can).