If $X\subset \mathbb P^{n}$ is a smooth (complex) hypersurface, one can compute its topological Euler characteristic $\chi(X)$ by taking the degree of the $0$-cycle $c_{n-1}(T_X)\cap [X]$.
If $X$ is singular, I do not know what can be said. Assuming this is a hard problem (but please, tell me if it isn't!), I am interested in the following particular case: $$ X\subset \mathbb P^{n} \textrm{ is a degeneracy locus.} $$ By this I mean $X=V(\wedge^{n}A)$ is the zero locus of a polynomial $\wedge^{n}A\in S=\mathbb C[x_0,\dots,x_{n}]$ which is the determinant of a square matrix $A\in M_{n-1}(S)$. I believe the singular locus of $X$ is $V(\wedge^{n-1}A)$, the locus of points where the rank of the matrix becomes $\leq n-2$. But this does not really help me. Do you know how to tackle this problem?
For a singular compact variety, there is an analogue of the Chern class $c_{n-1}(TX)$ which is called the Chern-Schwartz-MacPherson class $c_{SM}(X)$ which is either a class in the direct sum of homology groups or in the direct sum of Chow groups of $X$ and which satisfies $$ \int_X c_{SM}(X)=\chi(X) $$ (i.e. the component of degree 0 of the $c_{SM}(X)$ captures the Euler characteristic)
Here is a survey on this construction, and here is a paper describing an algorithm using this formula to compute $\chi(X)$, which the author M. Helmer has apparently implemented in Sage. Since you are actually interested in hypersurfaces, the earlier paper of Aluffi, thm 2.1, may be sufficient.