PC12 pilot reported a loss of aircraft control during climb after entering cumulonimbus clouds and IMC conditions prior to receiving IFR clearance. Pilot encountered extreme turbulence and moderate icing conditions after obtaining IFR clearance and continued to destination after exiting the weather.

Narrative

I was the sole pilot of a Pilatus PC-12. The day started with a trip from ZZZ1 to ZZZ to drop off the owner of the aircraft and passengers. Shortly after dropping off the passengers; me and one other occupant loaded up to fly the aircraft back to ZZZ1. While I was taxiing to the runway; I heard a couple of other aircraft call for taxi behind us. A few other times that I have operated out of ZZZ; tower had asked if we could depart VFR so we would not delay all of the other jets on the departure. In order to not make all of the other planes wait for us to reach a certain point; I offered to do the same this time and get my clearance in the air. On the way in; there were scattered clouds; but no actual ceiling. I made a departure off of [runway] XX with a left turn into the downwind to head east. I initiated a climb to 17;500 MSL. I was advised by tower that I could change frequencies; so I switched to Center. I could hear other aircraft calling center; but I could not hear centers response. As I climbed; the clouds were building and building quickly. I found myself in a position that was going to take me IMC. I advised center; center responded with a squawk code and then I didn't hear anything else. I put in the code and never heard 'radar contact'. I tried for several minutes to reach center. I was socked in and went IMC before I was able to re-establish contact with Center. I had no room left to try and maintain VFR.I finally got a hold of Center and it was a different voice. He said that I was told they were too busy to give me clearance; to maintain VFR; and that my ride would have been smoother if I had not departed VFR. I told him that I was trying to not delay the aircraft behind me and that it was not for my benefit. He came back and gave me a clearance to proceed direct ZZZ2 and climb and maintain FL210 I believe initially. During the climb I was in some cumulonimbus clouds and the turbulence was extreme. The VSI was going from a 2700 ft. a minute climb to a 700 ft. a minute descent. We were also picking up moderate icing on the wing that covered the boots in a matter of seconds and spread down the top of the wing itself. I could not see the bottom of the wing; but I assume with the pitch that the turbulence was putting us into; there was icing on the bottom of the wing as well. I was flying at 145 KIAS which is about 10 knots over the minimum climb speed in icing conditions. I tried to bring the nose down a bit more to 150 KIAS to give more of a buffer but the climb rate stopped. I brought it back down and that is when the roller coaster began of up and down. At one point; the pitch up was so abrupt from an updraft that the stall warning went off and disconnected the autopilot. The indication went away as I pushed the nose down and the aircraft's pusher system did not activate nor did the plane actually stall. I was able to maintain control of the aircraft and continue the climb out. All anti and de-icing equipment was on and was doing its best to keep the plane free of icing. After about 15 minutes we were able to climb out of and away from all of the precipitation and the ice started to clear from the wings. There was a slight vibration coming from the propeller as I suspected that ice had accumulated despite the prop heat being on. After several minutes of letting the heat run and being out of the precipitation; I saw a chunk of ice fly off of the prop over the front windshield. The vibrations stopped after that. The flight was then continued to ZZZ1 without any further issues. Knowing how the weather can be in the afternoon over this area; I will never again depart from one of those airports VFR in hopes of getting a clearance after departure. I will stick to the original plan and fly the IFR clearance that I was given on the ground. I will also do a better job at looking at the weather; knowing that despite being on the ground for a short time; and just having came from that direction; that things change quickly in that region.