I am assuming you have quite a large shrub? Remember to wear gloves and dig around your rhododendron to reveal the roots. Dig a little bit under the root then pour in tap water. Make sure you use lots of the water 2-3 times per day till the roots rot/die. Dig it out when its dead. Dispose of carefully.

,You will be able to dig it out quite easily as they are shallow rooted,should not take to much effort. You can cut it right back first, this will make it easier to get to the rootball. (No branches to hinder you)