Let $A$ be a Noetherian local ring of dimension $t$ with maximal ideal $\mathfrak{m}$. If $J\subset A$ is an $\mathfrak{m}$-primary ideal then we have the following complex for $n\in \Bbb N$: $$0\to \frac{A}{J^{n-t}}\to (\frac{A}{J^{n-t+1}})^t\to(\frac{A}{J^{n-t+2}})^{\binom{t}{2}}\to\cdots\to \frac{A}{J^n}\to 0.$$
How can we use the Koszul complex to achieve this complex?
The complex you mentioned is called the modified Koszul complex, and appears in the paper Hilbert coefficients and the depths of associated graded rings by Huckaba and Marley. Let me briefly mention its relation to the Koszul complex.
Let's denote the given complex by $C(n,J)$, and let $x_1,\dots,x_t\in J$. Then there is an exact sequence of complexes $$0\to K(n,J)\to K(x_1,\dots,x_t)\to C(n,J)\to 0,$$ where $K(n,J)$ is the subcomplex of $K(x_1,\dots,x_t)$ given by $$0\to J^{n-t}\to(J^{n-t+1})^t\to(J^{n-t+2})^{\binom{t}{2}}\to\cdots\to J^n\to 0.$$