I (only) slept around three hours because I prioritized doing my homework, which took me up to five hours to finish.

The "only" is not necessary, but it puts emphasis on the fact that you didn't sleep very much. Just a recommendation (:

I'm going to take my photography class after that.

"To take" is usually another way to say "to register" or to "be signed up for." For example, "I'm taking an English class this semester" or "I'm going to take a French class next semester." If this the case, this I would specify what "that" is referring to (:
However, I did want to ask to make sure. Is this what you meant to say or did you mean something like, "I'm going to go to my photography class after [you study with your friend]?"

I only slept around three hours because I prioritized doing my homework, which took me up to five hours to finish.

I sleponly got around three hours because I prioritized doingof sleep last night because of all my homework, which took me up tonearly five hours to finishcomplete.

The sentence is technically correct, but I see the focus of the sentence as "I slept around three hours" and so the bit at the end, "which took me five hours to finish", feels awkward to me. My suggestion here of "I only got around three hours of sleep last night" uses "only" to add emphasis on how little sleep that is.

Reading that, I now wonder "what could have happened?" What happened is "because of all my homework".

So my mind asks "was it really so bad?" In response: "which took me nearly five hours to complete." Again, "nearly" is for emphasis. "Up to five hours" feels like a very matter-of-fact way of saying it.

I'm going to take my photography class after that.

After what?

I'm going to take (my / a) photography class after that.

Did you finish studying French? If so, then "I will be taking a photography class soon."

If you haven't finished, change the above sentence to "Now, I'm studying French with my friend Ximena." and keep this sentence the same.