To get only one edge between each vertex, you only want a triangular matrix.
If you try something like this:
table1 = (table2 + table2') - triu((table2 + table2'))
table1 =
0 0 0 0 0
4 0 0 0 0
5 6 0 0 0
2 3 4 0 0
8 6 7 5 0
bg = biograph(table1,[],'ShowArrows','off','ShowWeights','on');
h = view(bg);
Now I have assumed you want to sum up the weights from both edges. I.e. the weight of the edge between 1-2 equal the weight of the edge from 1->2 + 2->1, from your original matrix.