Can I pilot a plane while having a chronic mental health condition?
You have a couple of options here:

As Skip Miller said, you can take flight lessons. Up until you want to solo, you don't need a medical certificate. Often flight schools have "Discovery Flights" for a reduced cost that still allows you to fly left seat and get half an hour to an hour of flight. Careful though, its addicting. 

Actual costs for this range from \$75 to over \$300 depending on the type of aircraft you are going to fly. Hourly rates for a Cessna 152 average around \$90 an hour, to a 172 up to \$150 an hour, and more sleek aircraft like a Cirrus SR-22 at \$300 per hour or more. Instructor rates go from \$35 per hour to \$65 per hour (typically) depending on the type of instruction.

Another option is to find a pilot who wants to build hours and go flying. They may let you fly the airplane around a little bit, but you'll most likely be flying right seat and probably won't let you do the more complicated phases of flight like take-off/landing and operations in controlled airspace. If you find a pilot willing to do this then you can share costs (split down the middle). 

By the way don't get discouraged about your medical condition if you really want to fly. There are also a few options there too. First, you could go through the rather lengthy process with the FAA to have your case reviewed and if possible issued a special issuance medical certificate. Some of what you list are automatically disqualifying depending on the medications you may take, but the AOPA has a good read on mental health and getting medicals issued.

Second, you could get a sport pilot license, which also does not require a medical (just a drivers license). There may be hiccups with your conditions here too, so its best to talk to an AME or flight instructor before getting too deep.



Let me answer your specific question:


  Is there a way I can fly with a copilot without a pilot's license?


I take this to mean that the "copilot" is a non-pilot rated passenger. Unless you are a certificated pilot, then no. A "co-pilot" must be just that, a pilot. Otherwise they are just a passenger sitting in the front seat. Since that person is a passenger, you must be a fully licensed pilot to fly with them (students cannot take passengers). 

If the copilot is a pilot rated individual, then yes, you can go flying with them (provided it is not a Part 135 flight). That person must be rated for the aircraft that you are flying and must meet currency requirements to carry passengers (you). You as a passenger are allowed to manipulate the controls under the supervision of the pilot provided the pilot is pilot in command and unless the pilot is a CFI, they are not providing instruction.