(My notations and definitions are similar to those in Hatcher's book.)
In Hatcher's AT, we construct a CW complex in an inductive way. This creates a new topological space.
Why don't we do a converse thing? Let $X$ be an already-existing topological space. Now I'm trying to give a CW complex structure on $X$ as follows, with some assumptions on $X$.
Assume there are index sets $\Lambda_1$, $\Lambda_2$, $\cdots$. For each $n=1$, $2$, $3$, $\cdots$ and each $\alpha \in \Lambda_n$, there is an $n$-cell $e^n_\alpha$ in $X$. Let $X^0$ be a discrete subspace in $X$. Let $X^n = \left( \bigcup_{\alpha \in \Lambda_n} e^n_\alpha \right) \cup X^{n-1}$. At the same time we have a continuous map $\phi^n_\alpha : D^n_\alpha \to X$ such that $\phi^n_\alpha (\partial D^n_\alpha) \subset X^{n-1}$ and $\phi^n_\alpha (\textbf{int}D^n_\alpha ) = e^n_\alpha$. Finally, assume the topology on $X$ is the coherent topology of its subspaces $X^n$. These are the assumptions.
Now is it enough to regard $X$ as a CW complex? Or, more assumption needed?
Not sure about your description since you said you wanted to start from a space, not to build one. The skeleta $X^0, X^1,..$ should be part of the data. You should look at proposition A.2 in Hatcher's to see the correct statement of what you are trying to set up. Essentially you start with a collection of maps $\Phi_\alpha: D_\alpha^n \to X$ and it gives (sufficient and necessary) conditions for these maps to be the characteristic maps of a CW-structure on $X$.
An alternative point of view (at least for the finite case) is the following: a finite CW-complex $X$ is a Hausdorff space together with a finite sequence of closed subsets $X^0 \subset X^1 \subset ... \subset X^n=X$ such that: