Learn Spanish (Latin America): Tengo que limpiar la cocina hoy.

You must include que! In Spanish, to say you "have to" do something, you always use the formula tener que followed by an action word in its basic form (like limpiar). If you leave out que, it sounds like "I have clean" instead of "I have to clean."

Doesn't que usually mean "what" or "that"?

Yes, que often means "that," and qué (with an accent mark) means "what." But when it is combined with tener to make the phrase tener que, it translates to the "to" in "have to." It is best to memorize tener que as a single fixed expression.

Since I am the one cleaning, why don't I change limpiar to limpio?

When you use a phrase like tener que plus an action, only the first verb changes to match the person doing it. So, tener becomes tengo for "I", but the second verb, limpiar, stays in its basic dictionary form (the infinitive).

I always forget how to say words starting with "h". Is the "h" in hoy pronounced?

No, the letter "h" is always silent in Spanish! You pronounce hoy exactly like the English word "oy" (rhymes with "boy").