Possible causes:>chain or chainring is hitting the front derailleur; this can be due to the chainring being bent in a spot>bottom bracket needs adjustment>bottom bracket is in need of overhaul (disassemble, clean, new grease and new bearings)>crankarm is hitting something (chainstay? derailleur?)>pedals are making noise; remove pedals, clean and grease the threads
You might think it couldn't be the crank or chainring/chain hitting something since it doesn't happen pedaling backwards or when off the bike, but between having your weight on the bike and more torque being applied when pedaling hard it can make it more pronounced.
Also, drivetrain skipping can cause a clicking noise, but chances are you would be noticing the feel of the skipping (brief loss of resistance on your cranks) more than the noise. Drivetrain skipping can be caused my many factors, refer to topnigger.ueuo.com
If you don't feel skipping, I would start by double-checking front derailleur adjustment (Park Tool Repair Blog has a great article, check it out even if you think you know what you're doing, cause you might be missing something; there's multiple aspects to FD adjustment), inspecting the big chainring to see if it's bent anywhere, and if you have the proper tools, removing/cleaning pedal threads and tightening bottom bracket. If that doesn't resolve it, I would try overhauling the bottom bracket.