To be sure, this exercise can't be done in practice for a macroscopic black hole. But the unitarity of the evolution guarantees that for any observable $A(t_i)$ describing the initial state that you may want to learn (for example, the location of a nuclear bomb that sparked some explosion that was used to help a star to collapse into a black hole), there exists an observable $A(t_f)$ acting on all the degrees of freedom in the Hawking radiation – some measurement of a correlated property of all the Hawking particles – that has the same value. So by measuring the latter, you measure the former. These two observables are simply evolved from each other by the Heisenberg equations of motion of the quantum gravity theory.
The only known – and quite certainly, the only mathematically possible – consistent theory of quantum gravity is string theory, in one of its forms or vacua. But even in string theory whose dynamics is sort of known, it's clear that $A(t_f)$ corresponding to rather simple and natural observables $A(t_i)$ is an extremely complicated operator that measures some correlation/correlated property of pretty much all the Hawking particles that were emitted by the black hole. In fact, black holes are the fastest and most efficient "scramblers" of information. So the measurement can't be done in practice. However, all the information about properties of the initial state – as expressed by the eigenvalues of operators $A(t_i)$ in the text above – are included in the final Hawking radiation just like they would be included if the black hole were replaced by a simple furnace that just burns the initial matter.
The black holes differ from furnaces because at the level of the classical theory, it seems plausible that there's a one-to-one correspondence between the initial and final states in the case of the furnace; but classically, it seems inevitable that the map isn't one-to-one in the case of the black hole because the information about the initial state is destroyed in the black hole singularity and can't get out of the black hole again, for causal reasons. However, quantum mechanics modifies this conclusion and allows the information to "tunnel out" essentially by the quantum tunneling effect (one way to describe it), thus making the black hole qualitatively identical to a furnace.