Rayon is easy to dye like cotton, using a good tie-dyeing kit. (See "How to Dye Rayon".) However, polyester cannot be dyed with any dye that will work on rayon. Polyester can be dyed only with a special kind of dye developed solely for synthetic fibers, which is called disperse dye. (See "Dyeing Polyester with Disperse Dyes".) If you want to dye both layers, perhaps you should use fabric paint or dye transfers, instead of regular dyeing.

If the original color is too dark or intense, it will be difficult to color it. Both dyes and most fabric paints are transparent and will not show up on top of a dark color. I don't recommend that you use bleach on polyester, though. Chlorine bleach can permanently damage polyester, turning it to a dingy yellow color which cannot be removed. Instead, I recommend that you buy a sulfur-based dye remover. (See "What chemicals can be used to remove dye?".) Although I don't usually recommend Rit all-purpose dye, I do strongly recommend another Rit product, Rit Color Remover. It works best when used on the stovetop with added heat, but it's easiest to use in the washing machine with hot water. I'd recommend trying the washing machine method first, because it's so much easier, and you can always repeat with the stovetop method if necessary. Depending on the size of your washing machine, you will need two or three boxes of Color Remover, and you should turn up your water heater to the highest safe temperature, if you're comfortable with adjusting the water heater.

I have to warn you that there's no guarantee that trying to remove the original color will work. Some dyes that are used in manufacturing clothing are easily removed; some are partially removable, or turn a funny color; and some will not go away no matter how you treat them. Since we don't know what dyes were used to color your dress teal in the first place, there's no predicting. It is worth a try, though. It works more often than it fails.

After you've removed as much color as you can, the question is, what should you use to tie-dye the dress? If none of the color was removed by the Rit Color Remover, then you can only chose an opaque fabric paint, such as Neopaque, but you will have to apply it thickly enough that you'd be able to feel the paint on the fabric after it is dry. Let's hope that this is not necessary.

Dyeing the rayon part with a Procion MX tie-dye kit, such as the Jacquard tie-dyeing kit, will work very well, but it won't color the polyester at all. (Don't use the Rit tie-dyeing kit, because it is not as good.) If it is important to you to dye both layers, this will require two stages of tie-dyeing, in which you first use Procion MX to tie-dye the rayon, wash the dress, and then use disperse dye to tie-dye the polyester, but this is almost certainly more trouble than you want to go to, unless you have a clear vision and a strong need to create it. 

The simplest solution would be to substitute a very thin fabric paint for the dye (but then skip the soda ash that's required for Procion Mx dyes). The results are not quite the same as tie-dyeing, but close enough, and some fabric paints work on both natural and synthetic fibers. I recommend either Dye-Na-Flow or Dharma Pigment Dyes for this purpose, diluted with the maximum amount of water recommended in the instructions. (See "Fabric Paints: a different way to color fibers".)

One last alternative would be to tie the dress and then drop it into a boiling dyebath that contains both "iDye" direct dye and "iDye Poly" disperse dye. With immersion tie-dyeing, you're pretty much limited to shades of a single color, unless you want to use the very unpleasant method of holding the garment only partially submerged in the boiling dye mixture, using a pair of tongs,for half an hour per color. The good thing about this method is that the "iDye" dyes the rayon or cotton at the same time that the "iDye Poly" dyes the polyester.