Hitting out the rough around the green requires a few different skills and lots of reps. The first skill is to assess the lie so that you can adjust technique accordingly. Secondly, good stability is required to be able to rotate and control the golf club. Lastly is to have the ability to adjust attack angles through proper wrist hinge as dictated by the lie.
To improve the first skill, make sure you practice all the different lies around the green and take the time to break down how the ball is sitting in the grass. Is there a lot of grass behind the ball? Is the ball sitting down so much that the club will bounce off the ground? Being able to assess these factors will help you to adjust technique accordingly.
Secondly, being stable requires good function. Yes, function dictates form even in chipping! Having a good stance width in the rough will allow for good rotation and control of the club. Somewhere around hip width will usually provide a stable base and the ability to move well.
Lastly, using the proper amount of wrist hinge to adjust for the lie will allow for the best contact possible and ultimately ball control. If the ball is sitting on top of the grass then a shallow attack angle is required, which means less wrist hinge back and down. When the ball is sitting down a more abrupt hinging of the wrist will allow for the proper attack angle so that there is not too much grass between club and ball.
Take the time to experiment with different lies asking these questions and then adjusting as you see fit. You will continue to learn with every rep.