Air carrier Captain reported track; airspeed; and altitude control difficulties when they flew into a thunderstorm departing MIA.

Narrative

On the SID climbing to 7000 ft we were on course to enter a thunderstorm. Approach tried to give us a heading 320 away from the storm; but on that heading we entered the thunderstorm. Upon reaching 7000 ft I leveled off however the aircraft continued to climb past 7000 ft. I was in the flight director and leveled off but because of the thunderstorm we were in an updraft and I had to continued to pitch down and reduce power. Upon reaching 7300 ft we had a TA caution message and ATC told us to stop our climb. At this point we were on a heading 5 degrees left and broke out and able to descend back down to 7000 ft. The aircraft accelerated up to 265 knots from 250 knots and the throttle significantly reduced. I was pilot flying and did not have the autopilot on at this time. Corrective action would be to engage autopilot early to reduce workload and request vectors earlier to stay out of the thunderstorm.There were thunderstorms in the vicinity. on arrival to the airport we were able to stay clear but on departure we went right through the thunderstorm. It was very congested frequently and it was very difficult to get vectors away from the storm.