First, let me say that I'm sorry that you are going through this. Dealing with a loved one who has an addiction can be painful and frustrating.
In terms of how to talk to them about it, that is a tricky situation. As a former addict myself, I can tell you that your loved one probably already knows that they are causing a lot of pain in your family. Chances are they also feel a lot of guilt and shame over their inability to control their addiction. Talking to them will bring these uncomfortable feelings to the surface. As a result, they may react with anger as a defense mechanism against feeling those painful emotions. If they do, try not to take it personally. Chances are they are angry with themselves and their addiction -- not with you or your family.
Here are some suggestions on how to broach the subject with them:
1. Find a time to talk to them when they are sober. Trying to talk to them while they are under the influence is a bad idea since they aren't thinking clearly. Consider finding a time when they are hungover or feeling remorseful over something they have done while drunk or high. This can make them more receptive to the thought of giving up their addiction. 
2. Avoid blaming them for their behavior. Remember that addiction is a disease -- not a conscious choice. You wouldn't blame them if they had heart disease or diabetes. The same should hold true for addiction. To keep them from feeling like they are under attack, stick with "I" phrases. For instance, try starting your sentences with phrases like "I noticed" or "I am worried".  Discuss any ways that their behavior is affecting your family. Be ready to provide solid examples of how their behavior is affecting you if they try to claim that you are overreacting.
Also, make sure that you are having a two-way dialogue rather than just talking at them. Ask questions that they have to answer with more than just a "yes" or "no" so that they feel like they are a part of the conversation rather than feeling like they are being lectured. This video has some excellent suggestions on how to talk to an addict in a way that they will understand:
3. Make sure that they understand how much you care about them. Let them know that you are bringing up their addiction because you care deeply about their well-being and are worried about them.
4. Discuss possible solutions to the problem. However, don't expect immediate changes. When someone is in the throes of addiction, they may not physically have the willpower to avoid drinking or doing drugs. Even if they desperately want to stop, they may not be able to without help. Depending on how severe their addiction is, they also may have to undergo a medically supervised detox program to avoid serious complications that can arise from suddenly stopping drugs or alcohol.  Before you talk to the person, come up with some solutions that you can recommend. For instance, consider finding information on outpatient treatment programs in your area or offer to go to counseling with them. Giving them options may be just the lifeline that they need to get help.
5. Even if they don't respond well to your discussion, let them know that you are available to talk anytime they are ready. While they may not be receptive to your message now, things may change in a few weeks or months as they continue to struggle with their addiction. Knowing that they can come to you to talk without judgment can give them a way out if they start feeling overwhelmed in the future.
6. Consider getting help for yourself and your family. While you may not be able to control your loved one's addiction, you can control how you personally react to it. Consider attending counseling or reaching out to groups such as Al-Anon for emotional support. Many support groups also offer online meetings if you or your family members are unable to attend in person.  Oftentimes, addictions are harder on family members than they are on the addict themselves. Don't be afraid to seek help in dealing with your emotions. Encourage your other family members to do the same. Together, by getting yourselves back on solid footing, you will be in a much better position to help the family member who is struggling with addiction.