Let $V$ be some complex irreducible representation of $SU(N)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of the Cartan matrix. This makes sense, since weights of an irreducible representation must differ by roots.
My question is, which rows of the Cartan are the correct ones to use? For instance, for $SU(5)$, suppose the highest weight is $(0,0,0,1)$. I know the Cartan matrix is
$$\begin{pmatrix} 2 & -1\\ -1 & 2 & -1\\ & -1 & 2 & -1\\ & & -1 & 2 \end{pmatrix}$$ Which rows do I subtract from the highest weight Dynkin label to obtain the other Dynkin labels, and when do I stop?
The rows of the Cartan matrix constitute the roots $\alpha_i$ in the Dynkin basis, where $i$ runs from 1 to 4 in this case, $SU(5)$. Notice that the weight also has 4 components (Dynkin labels).
By convention, label the components of the weight $a_i$ (annoyingly similar to $\alpha_i$). If $a_i$ is greater than zero, we subtract the root $\alpha_i$. So for every positive component of your weight, you subtract the corresponding root.
This is better understood using your example. The fourth component of the weight is the only one which is positive, therefore we subtract the fourth root. Explicitly, the weight is: $a_1 = a_2 = a_3 = 0, a_4 = 1$ so we perform $\Lambda_{highest} - \alpha_4$. This gives us $(0, 0, 1, -1)$, a new weight. We then continue the process by seeing that this time the third component is the only positive component, so we subtract $\alpha_3$.
If there are multiple components of the weight which are positive, we must subtract multiple roots and hence get two different new weights, each of which has the same "height". For example, starting with $(1, 0, 0, 1)$, which could be a weight in a different representation of $SU(5)$, we would need to subtract both $\alpha_1$ and $\alpha_4$.
Look up Slansky's "Group Theory for Unified Model Building" for a physics-angled encyclopedic approach to these calculations.