I've spent the last three summers working in Algonquin Park and will do so again this summer.
The old railbed on Mizzy Lake Trail is by far your best bet for birds. There is an open pine stand on the right side of Arowhon road as you drive to the railbed that can also be very good for boreal species.
Species:
Boreal Chickadee: Mizzy Lake Railbed, Arowhon Road Pine Stand, Spruce Bog or Bat Lake Trail. Should have no problem finding if you know the calls and put the time in. especially later in the summer.
Gray Jay: Best towards the end of the summer, I have barely ever seen them in June/July. Mizzy Railbed, Arowhon Pine Stand, Spruce Bog, Bat Lake
Spruce Grouse: Mizzy Railbed, Arowhon Pine Stand, Spruce Bog. Hard to look for, you have to get lucky.
Ruffed Grouse: common but hard to target. Walking through deciduous forest away from trails or first thing in the morning before other people have walked them is your best bet.
Crossbills and Pine Siskin: very erratic and hard to target, know the flight calls and you may see some flying by.
Pine Grosbeak: winter only
Black-backed Woodpecker: Mizzy Railbed or Arowhon Pine Stand. Not very reliable, probably easier in June.
Three-toed Woodpecker: very rare in Algonquin and almost exclusively in winter
Olive-sided Flycatcher: all over in appropriate habitat (e.g Mizzy Railbed) but hard to see later in the summer.
Of course you want to get out early in the morning, and you won't see much of anything midday.
Most of these birds are apparently much easier if you go a few hours further north.