When selecting a board to make an unbacked selfbow from, you need to look for three things. First is straight grain. The second, and most important thing, is that you need straight grain. Finally, you also need straight grain.
I'm not trying to mess with you here. Straight grain sounds so simple...but it isn't. You cannot take any grain run off for granted. It must be straight on all four sides, from tip to tip. Straight grain does NOT mean the same as growth rings running from tip to tip.
Read the Traditional Bowyers Bible vol. 1 on wood selection, and Traditional Bowyer Bible vol. 2 chapter on 'Board bows'. It will help you tremendously.
There are other things to look for in boards, but that depends on the wood species. Straight grain, however, must always be your priority. Only then you can look for good earlywood/latewood ratio (if applicable), heartwood versus sapwood, specific gravity, (dis)coloration etc.