Question: I normally write using the *American* English forms, but for some particular words, I tend to naturally write it in the *British* manner (like with the word *favourite* rather than *favorite*).

Is it wrong to mix both styles? If not \*wrong\*, does it looks weird?

Answer: 
I think most people would recommend you stick to one style or the other. Why? Well, it doesn't matter too much, but if you mix styles the reader might notice! And that's bad—if they're noticing stuff like that, then they're paying attention to *how* you're writing rather than *what* you're writing. 

In other words, you're distracting the reader. You probably don't want that. You want them to focus on what you have to say, right? So unless you have a specific reason to mix styles, just stick to whichever style seems appropriate.
I know a German girl here in Florida, who obviously learned British English in school. But after living in Florida for over 10 years, she has an immaculate American accent. The problem is that she still uses "bloke" where Americans would use "guy," she says "you know 't I mean?" with a British intonation but American accent, she says "zed" instead of "zee" and says "shedule" instead of "skedule." You would think these innocuous little details that nobody really notices, but the fact is that a lot of people find her amusing and even laugh at her (even some of her friends.) I personally find the inconsistencies jarring - especially "bloke." Every time she says it, I feel like saying "don't say that!" hahaha