A simple approximation algorithm for the TSP
This is called an insertion heuristic. The general TSP cannot be approximated unless $P=NP$. The question is more interesting for the metric TSP. In that case, Rosenkrantz et al. (1974) show that your heuristic gives an $O(\log n)$ approximation.

You can also try to choose a good vertex $v$ to be inserted in each step, instead of just taking an arbitrary one. One variant is farthest insertion in which $v$ is chosen such that is has maximum distance from the current $H$. Another variant is nearest insertion in which $v$ is chosen such that it has minimum distance from the current $H$. Yet another possibility is cheapest insertion in which $v$ is chosen such that the new $H$ has minimum length. Rosenkrantz et al. show that both nearest and cheapest insertion give a $2-2/n$ approximation and this is tight.

The full reference is:
D. Rosenkrantz, R. Stearns, and P. Lewis II. An analysis of several heuristics for the traveling salesman problem. SIAM Journal on Computing, 6(3):563-581, 1977.