Choose an affine open $V \subset X$ such that there exists a presentation $$ 0 \to \mathcal{O}_V^{\oplus n} \xrightarrow{A} \mathcal{O}_V^{\oplus n} \to \mathcal{F} \to 0 $$ for some $n$ and some matrix $A$. The ideal we are going to blowup in is the product of the Fitting ideals $\text{Fit}_k(\mathcal{F})$ for $k \geq 0$. This makes sense because in the affine situation above we see that $\text{Fit}_k(\mathcal{F})|_V = \mathcal{O}_V$ for $k > n$. It is clear that this is a $U$-admissible blowing up. By Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals} we see that on $X'$ the ideals $\text{Fit}_k(\mathcal{F})$ are invertible. Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $\text{Fit}_k(\mathcal{F})$ is an invertible ideal for $k \geq 0$. If $E_k \subset X$ is the effective Cartier divisor defined by $\text{Fit}_k(\mathcal{F})$ for $k \geq 0$, then the effective Cartier divisors $D_k$ in the statement of the lemma will satisfy $$ E_k = D_{k + 1} + 2 D_{k + 2} + 3 D_{k + 3} + \ldots $$ This makes sense as the collection $D_k$ will be locally finite. Moreover, it uniquely determines the effective Cartier divisors $D_k$ hence it suffices to construct $D_k$ locally. \medskip\noindent Choose an affine open $V \subset X$ and presentation of $\mathcal{F}|_V$ as above. We will construct the divisors and filtrations by induction on the integer $n$ in the presentation. We set $D_k|_V = \emptyset$ for $k > n$ and we set $D_n|V = E_{n - 1}|_V$. After shrinking $V$ we may assume that $\text{Fit}_{n - 1}(\mathcal{F})|_V$ is generated by a single nonzerodivisor $f \in \Gamma(V, \mathcal{O}_V)$. Since $\text{Fit}_{n - 1}(\mathcal{F})|_V$ is the ideal generated by the entries of $A$, we see that there is a matrix $A'$ in $\Gamma(V, \mathcal{O}_V)$ such that $A = fA'$. Define $\mathcal{F}'$ on $V$ by the short exact sequence $$ 0 \to \mathcal{O}_V^{\oplus n} \xrightarrow{A'} \mathcal{O}_V^{\oplus n} \to \mathcal{F}' \to 0 $$ Since the entries of $A'$ generate the unit ideal in $\Gamma(V, \mathcal{O}_V)$ we see that $\mathcal{F}'$ locally on $V$ has a presentation with $n$ decreased by $1$, see Algebra, Lemma \ref{algebra-lemma-add-trivial-complex}. Further note that $f^{n - k}\text{Fit}_k(\mathcal{F}') = \text{Fit}_k(\mathcal{F})|_V$ for $k = 0, \ldots, n$. Hence $\text{Fit}_k(\mathcal{F}')$ is an invertible ideal for all $k$. We conclude by induction that there exist effective Cartier divisors $D'_k \subset V$ such that $\mathcal{F}'$ has two canonical filtrations as in the statement of the lemma. Then we set $D_k|_V = D'_k$ for $k = 1, \ldots, n - 1$. Observe that the equalities between effective Cartier divisors displayed above hold with these choices. Finally, we come to the construction of the filtrations. Namely, we have short exact sequences $$ 0 \to \mathcal{O}_{D_n \cap V}^{\oplus n} \to \mathcal{F} \to \mathcal{F}' \to 0 \quad\text{and}\quad 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{O}_{D_n \cap V}^{\oplus n} \to 0 $$ coming from the two factorizations $A = A'f = f A'$ of $A$. These sequences are canonical because in the first one the submodule is $\Ker(f : \mathcal{F} \to \mathcal{F})$ and in the second one the quotient module is $\Coker(f : \mathcal{F} \to \mathcal{F})$.