Captain reported smoke entered the aircraft during initial climb. The flight crew corrected the Air Condition Pack configuration which eliminated the smoke with no further issues.

Narrative

We returned to the aircraft for the 2nd leg of the day. The weather was moderate snow and the aircraft was covered with snow and ice;so we arranged for deicing. We followed aircraft deicing procedures along with company procedures. During type 1 application we noticed a flicker in the aircraft electrical system; but after a thorough inspection found everything to be normal. After Deicing on an FBO ramp we taxied to RWY XXL. All taxi checks and systems checked normal. We taxied with our supplemental ground wing heat on and planned for wing and cowl heat on departure. Power up and takeoff were normal. At about 1300 feet in a turn to a vectored heading from tower we heard the passengers complaining of smoke; and we were also seeing heavy smoke entering the cockpit from the vents. I did not want to enter IMC so I [advised ATC]; stopped my climb and tried to maintain visual contact with the airport in case we needed to land immediately. After ZZZ tower replied 'stand-by'; we quickly determined the source and type of smoke. Removing the Packs; stopped the smoke and we both agreed the smell was not electrical but due to the deicing fluid entering the ACM's (Air Cycle Machine). The Deice truck must have sprayed into our APU air inlets. We switched the bleed air to engines; reopened the packs and all was normal. I contacted tower ; explained the situation and canceled the [priority handling]. Upon our arrival into ZZZ1; we delayed switching our bleedair source from engines to APU until our landing was assured. We did not get any smoke out of the vents and taxied into the ramp.