Time is directly connected to distance as far as movement is concerned. Actually, movement (a change of distance) is measured with time, and there is no movement without time (or time without movement), unless one is able to travel with infinite velocity. So you can always translate any distance into respective time according to the formula:
This solves your paradox. Achilles does not travel all the sub-distances traveled by the tortoise one-by-one in consecutive units of time; he travels a number of them at a time, and that's what allows him to catch up the tortoise. They travel different numbers of distance units within the same time unit.
Take a look at this:
"During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise."
See how the word "time" is used ambiguously here? Is it the same period of time or a different one that take Achilles and tortoise to travel the sub-distances? When Achilles is finally close enough to tortoise, but still behind him, in the next time unit he will travel as much as the tortoise in the same time and then some more. In one time unit Achilles can travel the distance between them, and also overtake the tortoise. He does not need to walk with very short steps and follow the footprints of the tortoise (which is suggested by the way the paradox is formulated).
And also, you cannot talk about traveling a distance (movement) without considering time. As I said - there is no distance change without time. One requires the other.
And notice that the paradox is actually stated in terms of time. You cannot avoid it. Especially that the whole paradox is about racing :)