I would start with a list of the five-card Poker hands, like
Professor Brian Alspach's (you can read his article about how to do this yourself).
Obviously if the board is better than a pair, your hand will be also better. So you only have to worry about the case where there is exactly one pair on the board, or all five cards are different and there is no straight or flush.
Put aside straights and flushes for the moment. To improve a one pair board, one of the cards in your hand must be one of the 11 remaining cards that match one of the cards on the board, or you must have a pair in your hand. These probabilities are easy to compute. For a no-pair board, both cards in your hand must match one of the 15 remaining cards that match a card on the board, a pair in your hand won't work.
If you assume that none of these things happen, you can compute the chances of three flushes or four flushes on the board, and your chance of filling them with your hand. You have to do this separately for one-pair on the board, no-pair on the board but one pair when combined with your hand, and no-pair on the board and no-pair with your hand. This is pretty easy, because suits are independent of ranks.
Straights are your headache. There are lots of different possibilities of three and four card combinations that can turn into different numbers of straights.