CE-525 Captain reported encountering severe wake turbulence departing LEB airport.

Narrative

I was PIC and also pilot-flying for all legs of a trip. We boarded our passengers for the live leg (Part 135) to TEB. The SIC briefed the passengers and confirmed that they intended to hold the infant in their lap for the flight. The safety briefing included instructions on requirements for seat belt use.We were cleared to taxi to runway 25 and ran the taxi and before takeoff checklists as usual. We were departing under IFR and the current weather included snow flurries and a broken to overcast ceiling with clear skies above the cloud layer. I briefed the takeoff and departure with the SIC which included the usual go/no-go decision points and also that we would climb above the clouds and cancel IFR once in VMC above the clouds and proceed under VFR to our destination (TEB).Tower instructed us to line-up-and-wait on runway 25 and advised that we were waiting for Departure Control to accept us after the jet that had departed before us had reached 5000' MSL. During this time the SIC again asked the passengers if they had their seat belts fastened as instructed during the safety briefing. The SIC said that they confirmed that their seat belts were fastened.We were cleared for takeoff and to climb on runway heading to 5000' MSL. We then took off and began a normal departure climb. After takeoff the SIC began the 'After Takeoff' checklist. I was hand-flying the aircraft and then engaged the autopilot as we were climbing through approximately 1000' MSL. The SIC then said 'go right; turn right' and instructed me to fly through a hole in the cloud layer. He then called Departure Control and cancelled IFR and asked to proceed VFR to our destination at 12500'. We were still below the cloud layer at this point and I was confused by his instructions to make a right turn to fly through the hole as this was not what we briefed and there was no obvious reason for me to do this.I was still in the process of configuring the plane for the climb. The SIC again said to turn right so I moved the heading bug to initiate a right turn. Shortly thereafter we hit severe-extreme turbulence. This was one solid jolt where the plane was momentarily uncontrollable and everything in the cockpit (soda cans; phones; iPads; etc.) hit the ceiling. I believe I disconnected the autopilot shortly before we encountered the turbulence or shortly thereafter. I was attempting to regain control of the aircraft and we got an over speed alarm several times as we were leveling off and within 10-15 KIAS of Vmo which is 260 KIAS below 8000' MSL. Everything in the cabin that wasn't secure also hit the ceiling. This included the passengers as they had apparently either unfastened or loosened their seat belts even though the 'fasten seat belt' sign was still illuminated. Once I had regained control of the aircraft the SIC went into the cabin to check on the passengers and then to get things cleaned up.During the time the SIC was in the cabin I was configuring the aircraft for the remainder of the climb to 12500' MSL and direct ALB (as instructed by ATC).Upon returning to the cockpit the SIC said that the passengers had some bumps but were otherwise ok and uninjured. The SIC then asked why I didn't turn right immediately and fly through the hole in the clouds and I said that we had not briefed that and that I didn't see a good reason to do it immediately. We then had a brief heated discussion and I then said that I was not going to discuss it further until after we landed and that we were going to concentrate on flying the remainder of the flight.After the flight the SIC said he had received a call that the passengers were claiming they were injured. There was some discussion between them regarding the safety briefing; etc. and apparently the passengers later decided that they were fine and dropped any claim of injury.Initially we thought we may have encountered turbulence related to the clouds and snow showers but after some thought the SICsaid he believed we had encountered the previously departed aircraft's wake. This makes sense to me as well.After taking some time to reflect upon this situation; I believe that I reacted to the SIC command to fly through the hole in the clouds in error. I should have continued to fly the departure as briefed. The SIC has nearly 20;000 hours of pilot time and many more hours in the CE-525 than I do (I was below 100 hrs PIC in the CE-525 at the time of this flight and still on Initial Operating Experience and 'High-Minimums') and is more comfortable doing such maneuvers in the aircraft than I am. I believe the fact that he has much more experience played a role in me attempting to comply with his instructions. As PIC; I should have been firm in refusing to follow the SIC instructions to fly through the hole in the clouds and continued the departure as briefed. It isn't possible to know if this would have resulted in avoiding the severe-extreme turbulence encounter but I believe it would have been the correct action and likely would have avoided the aircraft over speed alarm.