Level change is more of ‘altitude intervene/command’ mode. Basically it would hold the set speed and then climb or ascent on that speed. Pumping the rest of the engine thrust in climbing to the target altitude.
It’s often used when Air Traffic Control sends an instruction to the aircraft during the approach, as it ignores all the presets from the flight management system with your vertical profile and restrictions.
The profile mode (next to the level change) is the one best used for climb. Note that the aircraft will hold altitude/speed restrictions regardless of the altitude target. (So it’s perfectly fine to set a target of 15,000 feet and witness the aircraft not going past 8,000 feet when an altitude restriction would be in place). After passing the restriction, it will automatically continue the climb to the target altitude.
Usually you won’t find many restrictions during climb though, they are more to seperate you from the terrain when approaching an airport.
Normal behaviour would be to set the target altitude. Hit profile climb. Watch the aircraft accelerate to 250 knots below 10,000 feet (it’s a standard speed restriction). It will then climb (pretty steep) towards the target altitude at speed 250 knots. After passing 10,000 feet, the aircraft will first accelerate (speed always comes before vertical speed), and then continue it’s climb on the accelerated speed towards the target altitude.