I have an assignment in which I am supposed to create drawings/illustrations in a similar style to the old Asterix and Obelix comic-book. Is anyone here that can point me to a video tutorial similar to that style?
I don't know how much a video tutorial would help you. If I'm honest, I see little value in watching someone else draw. Much better to just get into drawing the characters yourself, and practising the style.
It is entirely possible to get that hand drawn, and hand coloured look in Adobe Illustrator. However, you are probably going to need a graphics tablet to do it. I couldn't imagine doing this with a mouse. The graphics tablet I use is one of the cheaper Wacom Bamboo tablets.
What I suggest you do is download some Asterix images, and try drawing some of the characters yourself, so you can get a feel for the style. Use the comic image for tracing over with ink lines.
Set up layers in your Illustrator document with one layer for the ink, one for the colour, and one for the image to be traced. Set your brush tool to a small size, and switch on the pressure sensitivity for the size, so that you can get variable width strokes.
Later when you want to try your own creations, perhaps create a separate layer for your rough sketch. Then you can simply switch it off after you have drawn better quality ink lines on the ink layer.
My example here is a bit wonky, but I think I quite like it that way - the less than perfect look gives it that hand drawn feel. I'm pretty sure with a bit more care you could get better results. Also, I am no cartoonist, so I'm already at a disadvantage there.
My attempt is the one on the left. You can also see my layers arrangement - one for the ink, and the colour layer under that.
You can find easily hours of videos which show, how one draws a scene or some character in Asterix style. Start from this, nobody can make it better:
Unfortunately we see only the tool movements. What's missing is the model inside artist's head, where he takes the lines from - his ideas and style, the soul of the characters and the essentials of all in the environment. Videos show virtually nothing of it. Without it you can at best only copy some existing drawings if you want to keep the style.
Your only possiblity for quality results is to build your own model. Make copies and find yourself what's essential. You can succeed if pen and brush already obey your mind. This is a hopeless case, if they don't. In this case search for Asterix style clip art.
In some "Draw Asterix" tutorials the maker reveals some rules that he has found useful - such as starting forms and sizes for nose, eyes, ears, helmet, etc... They are surely useful, but you must develop such set for everything which you are going to draw several times.
Do not expect you can legally publish your work as you will. Unpublished personal work is possible, but prepare to get strict NO or a hefty bill, if you are trying to get a legal license for publishing or selling something which resembles things in Asterix cartoons or movies. No reply is legally the same as =NO!
NOTE: If you take only the style, not a single detail, a court of law can still see you are making a plagiate. Discuss with a proper lawyer.
ADD due comments:
Some of the available videos use painting software. Essentially they still make sketches and add new layers when previous phases are ok. It's like real cartoon drawing and painting, but in the computer.
Computers allow to some degree you to reuse already made parts. If you like, you can save separately everything you think reusable - be it a sketch, outlined version or fully colored version - only be sure, that you have some system how to find and see the files and do NOT delete underlying sketches nor outline-only versions, only hide them, when they are not needed.
In computer you can also scale, rotate and warp things to fit better in a reuse case. Color changes are very fast, too when compared to repainting.
If you're intent on a digital workflow, I'd suggest looking into Rebelle as a colouring tool - it's a digital watercolour painting tool which has amazing drips, drying and paper capillary action engines - might help a lot with this kind of thing.