Cantrips levels with your character, not with your class
While it never explicitly indicates this anywhere in basic or PHB, it only sets the level at which the cantrip increases in power. One of 5e's design goals and one of the things that mattered about 4e was that there should never be a time when a magical user is forced to resort to a weapon attack for which he is not fit. The Cantrips act as spells at will of 4e and, as such, level with each "level" at 5e to maintain their usefulness. Relating this to the character level means that players who multiclass or, for example, start with a free cantrip (such as High Elves) can still use those cantrips throughout the game.
A clarification was added to the Sage Advice Compendium.
If a character has levels in more than one class, do the character's cantrips scale with the character's level or with the level in a spell casting class? Cantrips scale with character level. For example, a barbarian 2 / cleric 3 throws a sacred flame as a level 5 character. (Compendium Sage Advice V_2.0; page 11)