I would go with a semi- or demi-permanent dye that closely resembles your natural color and dye your hair with that. Try to stay away from permanent dyes since they would damage your hair even more. Also, I've heard from others that ash browns do not turn out nicely on bleached hair (apparently they sometimes create a green-ish undertone) so try to get a shade that is labeled golden rather than ash.
When hair is bleached the pigment is removed and must be replaced. Best results would be to add gold, as in light golden brown, to fill.
A semi/demi would be a good choice as you are wanting to deposit only, however, will fade with every shampoo, probably more than you want as there is nothing for the color to hold on to.
Too many people give permanent color a bad wrap. It's not going to damage your hair any more than it's already damaged from the prior bleaching unless you are not applying it correctly.
Get yourself a semi permanent color as in the light golden brown I suggested earlier, and use it as a filler. After you complete this step follow it up with something more permanent, so it will actually penetrate inside the hairshaft as opposed to just staining the outer layer. (by this I mean no more than a 10 or 20 volume peroxide). Don't be discouraged as you most likely will still get more fadage than you wish, but keep in mind, until you get the color molecules back in the shaft to accept the color nothing will hold.
Hot water opens the cuticle, thus aiding in fadage, therefore, wait at least 24 hours after coloring(as this process also opens the cuticle) to shampoo and use tepid water when doing so.
Good luck...and if all else fails, see a professional.
In the past I used to go from brunette to blonde and back. When going back to brunette after having my hair bleached blonde, I would usually just use semipermanent color and always had good results. Keep in mind that bleached hair will often soak up a lot of color and may turn darker than expected, so go with a color that is slightly lighter than you are wanting. I'd usually go with the lightest brown for medium brown hair. I can even use a dark blonde box and get a pretty golden brown. I also hear many warnings about the ash colors, but I haven't had any issues with them and have used them several times. Just in case, stick with neutral or golden hues! Neutral colors should give you a nice color if you don't want a warm brown, and golden will give you a warmer color. I've also used permanent color after being blonde and was happy with both Garnier Nutrisse and the Perfect 10 dyes. These were very moisturizing and my hair felt silky after using them. If your hair is dry or damaged, I'd stick with the semipermanent dyes.