Additive closure of a category Given a category $\mathcal{C}$, there is a (I believe) well-known way to obtain an additive category from that, called the additive closure of $\mathcal{C}$ (see eg Bar-Natan's Khovanov’s homology for tangles and cobordisms, Definition 3.2): first one turns $\mathcal{C}$ into an Ab-enriched category by replacing the Hom sets by their $\mathbb{Z}$-linearisations and later one considers formal direct sums (coproducts) of elements of $\mathcal{C}$.
As everything in category theory, it is better to define things by their universal properties. The following is an attempt to give the universal property of this additive closure.

Claim: Let $\mathcal{C}$ be a category. There exists a unique (up to equivalence) additive category $\overline{\mathcal{C}}$ together with a fully faithful functor $i: \mathcal{C} \hookrightarrow  \overline{\mathcal{C}}$ such that every object of $\overline{\mathcal{C}}$ is isomorphic in a unique way to the coproduct of a finite family of objects in the image of $i$.

Is the previous statement the correct characterisation of the additive closure of $\mathcal{C}$? If not, what is its correct universal property?
 A: Your description works well enough, in that it clearly specifies $\overline{\mathcal C}$ essentially uniquely as the category of tuples $(c_1,\cdots,c_n)$ of objects from $\mathcal C$ in which morphisms are given by viewing such a tuple as the biproduct of its objects, taken in the $\mathbb Z$-linearization of $\mathcal C$. Note that we must be a little careful about the meaning of "unique", here, since for instance biproducts are commutative and associative up to isomorphism: it's a bit clearer to say that each object comes equipped with the structure maps making it a biproduct of some objects of $\mathcal C$'s linearization.
Another of describing this category is as follows:
$\mathcal C$ is an additive category admitting a fully faithful embedding of $\mathcal C$ such that, if $F:\mathcal C\to \mathcal D$ is any functor from $\mathcal C$ into (the underlying ordinary category of) an additive category $\mathcal D$, there is an essentially unique additive functor $\overline F:\overline{\mathcal C}\to \mathcal D$ supplied with a natural isomorphism $\overline F\circ i\cong F$.
This kind of characterization works much better for constructions like "the free completion of $\mathcal C$ under filtered colimits", in which objects do not have a unique-up-to-isomorphism decomposition as a filtered colimit of objects of $\mathcal C$. Such constructions arise often; for instance the category of abelian groups is the free cocompletion under filtered colimits of the category of finitely presentable abelian groups. An abelian group $A$ does have a canonical description as a filtered colimit of finitely presentable groups (namely $A$ is the colimit of the diagram of all maps to it from such groups) but this description is non-unique. For instance $A$ is also the filtered colimit of all its finitely-generated subgroups.
