We could take a look at the dyes available during the time of Joseph and then make a pretty good guess about the colors that could have been used in his coat...

It was red and yellow and green and brown
And scarlet and black and ochre and peach
And ruby and olive and violet and fawn
And lilac and gold and chocolate and mauve
And cream and crimson and silver and rose
And azure and lemon and russet and grey
And purple and white and pink and orange
And blue

- according to Andrew Lloyd Webber and Tim Rice (Joseph And The Amazing Technicolor Dreamcoat)