For clarity, I'll illustrate the differences between the three phrases you have quoted by means of a concrete example. Namely: the speaker [in all 3 phrases] is named Alice, and she's referring to herself and her brother Bob as us.
A picture of us
This phrase just means a picture which shows the individuals who are the antecedent of the pronoun us. Using our example, the picture shows Alice and Bob. As @Sam commented, there is no implied suggestion of who did or didn't take the picture. The picture could have been taken by Alice, Bob, or any other person.
A picture of ours
This (using "of ours" instead of "of us") conveys a significant difference in meaning - a picture of ours means a picture belonging to us. It does not have to depict "us"! So, the picture referred to by this phrase must belong to Alice and Bob, but doesn't necessarily show either of them: it might, for example, be a picture of their friend's cat. This phrase also doesn't give any information as to who took the picture.
A picture of ourselves
Unlike the other two, this one actually does carry the connotation that the picture both represents and was taken by the same individuals. This phrase is likely to occur in a context like "We took a picture of ourselves by the waterfall." So, the picture shows Alice and Bob, and was taken by either Alice or Bob. I only said either (and not both) of them because it isn't possible for two people to simultaneously take the same picture, as far as I know.