No such definition can be given for a general topological space, because a general topological space can have very few morphisms from or into $[0;1]$. E.g. consider $\Bbb Q$: there are no non-trivial continuous paths, containing only rational points. On the other side, Urysohn's lemma states that only normal topological spaces have enough continuous functions on them.
Your proposed axiomatization looks very similar to the axioms of $(\infty,1)$-categories. There you have objects, corresponding to points, morphisms between them, corresponding to paths, 2-morphisms between morphisms, corresponding to homotopies between paths etc. Every topological space has a canonically associated $(\infty,1)$-category, constructed just as above. There is an adjunction between the bicategories of topological spaces and of $(\infty,1)$-categories. In fact, it is even a Quillen equivalence (equivalence of homotopy categories) for some natural choices of model structure on categories (i.e. for some notion of homotopy equivalence). However, the image of this adjunction in $\mathcal{T}op$ consists not of all spaces, but only of CW-complexes. So the answer is: it works, but only for very nice spaces.
You can look further information on $(\infty,1)$-categories and quasicategories. A short survey of theory can be found in the first chapter of J. Lurie's "Higher topos theory".