In Schlichting's paper Negative K-theory of derived categories. Math. Z. 253, 97–134 (2006), a definition of negative $K$-theory of triangulated categories is given. These are abelian group valued functors $\mathbf{K}_{i}$ with $i\leq 0$.
These groups agree with the known negative $K$-theories, for example in the Thomason-Trobaugh paper mentioned by Prof. Grayson. However they use idempotents in an essential way.
In particular, if $E$ is an exact category then the bounded derived category $D^b(E)$ is triangulated and $\mathbf{K}_{0}(E)$ is defined to be the usual $K_0$ of $\widetilde{E}$, the idempotent completion of $E$.
If $E$ is idempotent complete, then the unbounded derived category $D(E)$ exists, and the $\mathbf{K}_{-1}(E)$ turns out to be the quotient of the abelian monoid (under direct sum) of isomorphism classes of idempotents in $D(E)$ by the submonoid of split idempotents. In particular $K_{-1}(E)=0$ if and only if $D(E)$ is idempotent complete.
More generally, if $A$ is an idempotent complete triangulated category then $\mathbf{K}_{-1}(A)$ is zero if and only if the Verdier quotient $B/A$ is idempotent complete for all full triangulated embeddings $A\to B$ with $B$ idempotent complete.
Incidentally, Schlichting conjectured that whenever $A$ is a small abelian category, then $\mathbf{K}_{i}(A)=0$ for all $i<0$.
This conjecture was generalised to stable $\infty$-categories by Antieau, Gepner, and Heller in K-theoretic obstructions to bounded t-structures. Invent. math. 216, 241–300 (2019), wherein they conjecture that $\mathbf{K}_{i}(A)=0$ for $i<0$ whenever $A$ is a stable $\infty$-category with a bounded $t$-structure.
Neeman gave a very simple and elegant counter-example to both these conjectures in a recent preprint https://arxiv.org/abs/2006.16536.