Your attempted solution apparently amounts to substituting the definition of addition and multiplication, but these are given to you so there's no need to "find them". In fact, there are lots of different ways to define addition and multiplication. Most of them would not result in a field (think of $(a,b)+(c,d) = (0,0)$). On the other hand, there might be more than one which will result in a field. So the question really depends on the definition of addition and multiplication.
In order to check whether this object is a field, you have to verify all the field axioms. For example, in a field $x + y = y + x$. In your case, we need to verify that $$(x_1,x_2) + (y_1,y_2) = (y_1,y_2) + (x_1,x_2).$$ All we need to do is substitute the definition of addition: $$(x_1,x_2) + (y_1,y_2) = (x_1+y_1,x_2+y_2)$$ whereas $$(y_1,y_2) + (x_1,x_2) = (y_1+x_1,y_2+x_2),$$ and both expressions are equal by the commutativity of real addition.
Some of the field axioms require you to identify a zero element and a unit element, i.e. the $0$ and $1$ of the field. What should these be? Do they satisfy all the required axioms? For example, is every non-zero element invertible?
For your purposes, the elements of $GF(256)$ are polynomials in $x$ of degree $\le 7$ with coefficients in $GF(2)$. $GF(2)$ is just $\{0,1\}$ with binary addition and multiplication, but there are no carries: 1+1=0 in $GF(2)$.
So for example, $x^4 + x^3 + 1$, $x^7 + x^2 + x$, $x^5 + x^3 + 1$, are all elements of $GF(256)$. There are 256 elements in all (hence the name $GF(256)$).
Addition of 2 polynomials in $GF(256)$ is straightforward. For example:
$(x^4 + x^3 + 1) + (x^3 + x^2 + 1) = x^4 + x^2$. This is just normal addition of polynomials, but the coefficients of the calculations take place in $GF(2)$. So when I added the 2 $x^3$ terms together, the coefficient became 1+1=0 (so the $x^3$ term disappeared altogether). As bit sequences this would be: 11001 + 1101 = 10100. This is straightforward to implement in most programming languages: It's just an XOR of the bit sequences.
To multiply 2 polynomials in $GF(256)$, first you multiply the polynomials just like ordinary polynomials but again, remembering that the calculations take place in $GF(2)$. Then divide the result by $x^8+x^4+x^3+x^2+1$ and take the remainder. Unlike addition, this is less straightforward to implement from scratch. (You can't use the usual multiplication operator in your programming language to do the multiplication, because ordinary multiplication has carries, but $GF(2)$ does not.)
See the example here which is a slightly different polynomial (they use $x^8+x^4+x^3+x+1$ instead of $x^8+x^4+x^3+x^2+1$), but it's the same process. There is also a description of a multiplication algorithm there (you'll have to change it slightly for your polynomial).
Best Answer
For the multiplication table, take a look at what happens when you multiply $x$ with itself. Why is that a problem?
As for the method of doing it -- I just remembered what a field is defined to be: a commutative ring with a multiplicative identity, no zero divisors, and every element has a multiplicative inverse. There are actually multiple reasons why this is not a field. Take a look at the $x$ row in the multiplication table. Do you see any $1$s belonging to that row? No. Why is that a problem?