"The" category of Turing machines?
If your objects are Turing machines, there are several reasonable possibilities for morphisms. For example:

1) Consider Turing machines as the automata they are, and consider the usual morphisms of automata (maps between the alphabets and the states that are consistent with one another) which also either preserve the motions of the tape head(s), or exactly reverse them (e.g. whenever the source TM goes left, the target TM goes right and vice versa).

2a) Consider simulations or bisimulations.

2b) Along similar lines, you can consider when one TM can be transformed (by a computable function) to simulate the other one. This can be done at the level of step-wise behavior, or as Yuval suggested in the comments, at the level of input-output, that is, a morphism from $T_1$ to $T_2$ (or maybe the other way around) is a computable $f$ such that $T_1(x) = T_2(f(x))$ for all $x$. 

3) Consider the transition graph of the Turing machine (each vertex is a complete description of the state of the machine and the tapes, with directed edges corresponding to the transitions the TM would make) and consider morphisms of graphs. For TMs this is a very coarse relationship, however, as it essentially ignores the local nature of computation (it ignores, for example, what the contents of the tapes are).

I think the real question is: what is it you want to know about TMs or to do with them? In the absence of this, it's hard to give arguments for any one definition over another, beyond naturality (in the usual sense of the word, not the categorical sense).