I got up at 5am in the morning. I got up at 5 in the morning.

"am" already implies morning so it's redundant. Or you can remove the second part and just write "I got up at 5 am."

I got up at 5am in the morning. I got up at 5 in the morning.

Or: "I got up at 5 am."

Today I got up at 5am in the morning. Today I got up at 5am.

You could either say "Today I woke up/got up at 5am" or "Today I woke up/got up at 5 in the morning." the "Today" is optional since the title does say what day you are writing about.

I stayed up late last night because I had some energy drinks, although I woke up nuraturally. I stayed up late last night because I had some energy drinks, although I woke up naturally.

Although I stayed up last night althoughdrinking energy drinks, I woke up nuraturally. Although I stayed up last night drinking energy drinks, I woke up naturally.

It is a little more natural for me to put the "although" at the front of this sentence. :)

Even though I stayed up last night although I woke up nuraturally. Even though I stayed up last night I woke up naturally.

Because I've just drinking some drinks.(combined with previous sentence) (combined with previous sentence)

based on the context given by your previous sentence it's not clear: did you stay up late because of these drinks, or were you woken up because you drank a lot before sleeping and woke up naturally from a full bladder? I will guess that you stayed up late as a side effect of having an energy drink, my correction is based on this guess.

Because I've just drinking some drinks.

I'm not sure what this is trying to convey. You want to connect it to the prior thought, so I might go with FM94's suggestion. I corrected it with that in mind and also combined the sentences. "Because I've just been drinking some drinks" = what kind of drinks? Alcoholic? Non-alcoholic? Caffeinated? In English, this sounds a bit like you're avoiding telling someone exactly what you've been drinking (because you don't want to tell them). "Because I've just been drinking" = implies that you were drinking alcoholic beverages, which doesn't make a whole lot of sense if you woke up naturally, since most people who drink alcoholic drinks tend to wake up with a hangover.

Because I've justI was up late because I was drinking some drinks. I was up late because I was drinking some drinks.

Just changed the sentence structure a bit. I didn't add in a type of drink because I am not sure if you mean coffee or energy drinks but i would recommend adding the drink type in to the sentence before the word "drinks".