Can I remove this mold by myself?
Yes that is mold and it's very common bathrooms.

Yes you can try to remove it yourself. 


Make a bleach solution with 1 tablespoon bleach and 1 cup warm water with 1-2 drops of soap. Dish washing soap is fine. 
Put on eye protection. 
Use an old toothbrush to scrub at the mold to remove it. The soap should help remove the mold and the bleach will help kill it but keep in mind mold often stains things. 


Source: I get this mold all the time in the bathroom and I've tried various ways to fix it. Sometimes I just have to remove the old caulk and add new caulk. The best way is prevention with good ventilation. 

Some more tips: 


If you or the landlord paints over this, add mold inhibitor to the paint. It will help on painted surfaces. 
I've found mold living on painted aluminum window frames. So there must be something there in the paint for it to eat. This mold can grow in unexpected places.
If you are a renter, this might be a health code violation and that means you can get the landlord to fix it. Look into a renter's association in your area for advice.
Or make a deal with the landlord. If he buys the paint with mold inhibitor, you can paint the bathroom for free.
You can buy buckets of moisture absorber, which is essentially calcium chloride, sold as non-salt ice melt in the winter. This stuff is also sold as moisture absorber at the Dollar Tree, but only in the small buckets. Online the big buckets are more expensive. DT link. Link to product image.