Let $X$ be an irreducible closed subscheme of dimension $1$ of $\mathbb{P}^n_k$ for some $n$. Let $X'$ be the closed reduced subscheme associated to it, which is an integral projective curve. Definining the degree of $X$ as the top degree coefficient of its Hilbert polynomial, I think that one should get an equality $deg(X)=m deg(X')$ where $m$ should be the length of $\mathcal{O}_{X,\eta}$ , being $\eta$ the generic point of $X$.
I've tried to write down the exact sequences relating $\mathcal{O}_X$ and $\mathcal{O}_X'$ but I did not get anything really useful.