How can we systematically convert a non-deterministic Turing machine into a deterministic Turing machine that recognizes the same language?
The deterministic machine simulates all possible computations of a nondeterministic machine, basically in parallel. Whenever there are two choices, the deterministic machine spawns two computations. This proces is sometimes called dovetailing. The tape of the deterministic simulator contains a list of configurations of the nondeterministic one, and performs a step on each of these in turn. This requires quite some administration, and the capability to move aroud data when one of the simulated configurations extends its allotted space.
a worthwhile question because the algorithm is not really so trivial to a neophyte and worth studying for its key/basic theoretical implications, and your question is not specifically limited to recursive languages! a key is to recognize the algorithm as a breadth first search of a (possibly infinite) graph of all possible transitions, ie exploration of all edges of a graph in parallel so to speak. (exercise: explain why the similar graph-traversal algorithm, depth first search cannot work.)
the construction is relevant to, and shows up in, the famous/fundamental Cook proof for the NP completeness of SAT. basically solutions of the SAT construction are 1-1 correspondence with NTM acceptance paths in P-time.
moreover there is significant theory (somewhat similar/analogous) of converting other nondeterministic machines to deterministic ones eg NFAs to DFAs. in general the complexity of the corresponding classes (deterministic vs nondeterministic ones) is open for many related questions.