Instead of adding up scores and figuring out the new averages, a quicker way (useful for the countdown round) is as follows: Since the average is 90, we first calculate how many points below 90 he already is: he's scored 87, 85, and 87, meaning that he is 3, 5, and 3 points below 90 respectively, making for a total of 11 points below. We then know that his remaining two tests must be a combined 11 points above 90, and since they differ by 3 points, we know that one test is 4 above 90, and the other is 7 above 90, meaning his highest score was $ \boxed{97} $.