Chat room style text animation with expressions

This expression will offset the height of the text layers above the first layer by 130% of the height of the previous text layer. All you have to do is set each text layer's in point. 

if (thisLayer.index !== 1 && time >= thisComp.layer(index -1).inPoint){
	refLyr = thisComp.layer(index -1);
	ofstsrc=refLyr.sourceRectAtTime();
	ofst = ofstSrc.height * 1.3;// set percentage of offset to 130%
	[refLyr.position[0], refLyr.position[1] - ofst];
}
else{
	value
}

It gets a little more complicated if you want to add a background layer to the text, but the basic expression would work for the position. You would have to change the index value to -2 and possibly use the shape layer background height instead of the sourceRectAtTime().height for the text layer.

This is what it would look like with an added opacity text animator added.

You could also create an ease(t, tMin, tMax, value1, value2) operator that works on time from the inPoint to ease the layers into their new position. 

Maybe this will get you started. I uploaded a sample comp.