Hi Framer,
Boeing's normal procedures dictates to set climb thrust at Thrust rerduction height (by pushing N1 button on the MCP) and verifying the thrust mode display have changed to CLB or R-CLB (whichever be the case) and set Flaps up speed at Accceleration height to retract the flaps on schedule. Once flaps and slats have been fully retracted, set VNAV by pushing it on the MCP.
Level Change is a Supplementary procedure and it is not contemplate on the Boeing's amplified normal procedures, however if your company's SOP dictates to use LVL CHG then you have to do it.
I remember some pilots to push the VNAV button at 1000' AGL, the airplane reduced to Climb thrust and set speed 250 KIAS all at once and start to command flap retraction immediately thereafter. But again, it was not a normal procedure nor adhered at any SOP.