Adding Halftone

Option drag the shape you want to add texture to.

Select the Gradient Tool and fill the shape with a gradient.

One end is 50% black with an opacity of 100%.

The other end is white with an opacity of 0%.

Go to Effect > Pixelate > Colour Halftone

Max Radius 6

It should look like this.

Go Object > Rasterize > Bitmap at 300ppi

Then Live Trace with these settings

Expand

recolour and drag back over the original shape.