Draw up an addition and a multiplication table for the factor ring 2Z/8Z.
Let $\displaystyle G=2\mathbb{Z}$ and $\displaystyle H=8\mathbb{Z}$.
Then $\displaystyle G/H = \{ H,2+H,4+H,6+H\}$.
Let $\displaystyle a=2+H$ and $\displaystyle e=H$.
Then $\displaystyle G/H = \{ e, a,2a,3a\}$.
Adding them is simple, for example, $\displaystyle 2a+3a = 5a=a$ by using $\displaystyle 4a=e$.
Multiplying is similar, $\displaystyle 2a\cdot 3a = 6a=2a$.
With those facts you can form a table.
The elements of 2Z are ..., -6, -4, -2, 0, 2, 4, 6, 8, ...
The elements of 8Z are ..., -42, -16, -8, -0, 8, 16, 24, ...
The elements of 2Z/8Z are 0, 2, 4, 6.
You can consider these as being 2x the elements in Z_4, i.e. 2 x {0, 1, 2, 3} and do all the work on Modulo 4 arithmetic and multiply everything by 2 when you've finished, or you can directly use modulo 8 arithmetic and create the multiplication and addition tables based on the defined subring of that.
As long as you know how to do modulo n arithmetic you're home.