First, please note the difference between もらう ("to receive") and くれる ("to give [me]; to do [for me]"). 贈ってくれる does not mean "I receive"; it means"[someone] gave me." This distinction makes it clear that the subject of this sentence is not "I," and therefore that it is not "I" who became good at drawing birds.
With that in mind, since ようになる implies a shift from one state to another, it should be evident that this sentence refers to more than one Christmas gift – that is, that once the person in question became proficient at drawing birds, it became his or her regular practice to give pictures of doves to the writer as Christmas gifts. [EDIT: The preceding sentence is actually incorrect; see Naruto's comment.] The lack of context in the original post makes it impossible to translate the sentence with any precision, but it might be something like "He [or she] got good at drawing birds, and even started giving me [or us] pictures of doves for Christmas" or "He [or she] got good at drawing birds, to the point that he [or she] even gave me [or us] a picture of a dove for Christmas." [EDIT: Possible translations modified in an attempt to bring them in line with Naruto's correction.]