Miller indices: How does one translates the orientation to the origin of the unit cell?
Let's say your structure contains a short piece of helical DNA, and it (the helix axis) starts at point P1 and goes to point P2. You want to know the Miller index associated with that direction. So you move the entire helix (without rotating) so that P1 is now at the origin. The new position of P2, lets call it P2(prime), will be at:

$$ P_2' = P_2 - P_1 $$

Just checking that is is a translation (I have to subtract the same vector from the entire structure, otherwise I'm rotating or warping it):

$$ P_1' = P_1 - P_1 = (0,0,0)$$

Again, the pair of two points still points in the same direction, but now $P_1$ is at the origin. Taking the fractional coordinates (i.e. along the unit cell axes) of $P_2'$, you can turn those into Miller indices $h$, $k$, and $l$ by multiplying with a large number and rounding to the next integer. Generally, the direction will be off because of the rounding. However, in special cases, e.g. if your DNA is on some crystallographic symmetry axis, the Miller indices will match your direction exactly.