The obvious thing to do is to choose $H$ a general hypersurface containing $X$.  In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.  

Certainly $H$ is smooth away from $X$ by Bertini.  I don't see why it should be smooth along $H$ though (unless of course, $X$ is a complete intersection).

In general, you are still ok *locally*, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a variety which is smooth near that point (they might not glue, or be smooth elsewhere though).  This follows from page 171 of Matsumura's commutative algebra.  See in particular 21.2(ii).