As you wrote, mental health struggles affect not only the person who is experiencing them but also family and friends that care about them. The fact that you are reaching out with your question illustrates just how much you care for the wellbeing of your relative and the desire to help.
Try to create a safe non-judgmental space in which they can talk openly and honestly about what they are experiencing. Ask them how they are feeling and be open about your own experiences with mental health. Experiencing a manic or depressive episode can be quite scary to the person who is going through it especially, when as you said, they have not sought out help as of yet. Although you may not be able to understand their experience just the fact that you are trying to understand speaks volumes.
Educate yourself as much as you can about things like Bipolar disorder, depression, and mania. There are books that have been written from the perspective of someone with Bipolar that can help provide insight into what the experience can be like. There are lots of resources online, but be sure that you are looking at well established sites such as official mental health agencies and governmental departments.
If you can learn what some of their triggers are, this can help a lot as well. If for example, there are certain behaviors you notice before a manic or depressive episode that you are aware of, you could gently let them know in a way that reflects care and concern. Although there are many things that may be a trigger some common triggers include sleep disturbances, physical illness, stressors related to work, relationships or money, the death of a loved one or the end of something (such as graduation). Of course, these examples are not all encompassing. Triggers are as individual as each person is.
Finding a balance between support and control can also be a bit of a tricky one. Often times, people with various mental health diagnoses experience overly controlling or critical behavior from friends and family. Ongoing communication and acceptance of one another's feelings is crucial for this. For example, talk with your loved one about how you feel in an open honest way at a time that is conducive for the conversation. If they are experiencing a low, that would not be a good time to talk about how you feel when they are depressed.
A good conversation to have with them when you are both in a good head space would be to ask them what they need when they are experiencing a low (or a high). They may say that they need company or conversely they may let you know that they wish to be left alone. Remember that ultimately, each person is an expert on their own experience and knows best what they need. As long as the person is safe, if they request to have time alone, for example, one way that you can show them that you are supportive is by honoring their need.
Remain calm and provide support whether you are faced with a manic episode or a depressive episode. For example, sometimes during a manic episode people may see or hear things that are not there. This is often a time where we ask the question of "is it better for me to disagree with them and potentially cause further upset or should I just agree and potentially further fuel the belief further?" Stay calm and reassure the person that you are aware that what they are experiencing feels real to them. It's often helpful to focus on what they are feeling rather than confirming or challenging their sense of reality, which in those moments feels incredibly real for them. By supporting the person you are helping them to feel seen and heard.
A good place to start as far as getting a diagnosis is with a Primary Care Physician or General Practitioner. If possible try and go with your relative, with their permission of course, to the appointment in which the subject will be discussed. It can sometimes be helpful to have examples of some of the concerning behaviors that you have seen, especially if the person doesn't believe that they are ill. The best combination of treatment typically involves medication, therapy, and self-management.
It goes without saying that while you are helping them and actively working on supporting them so that you can have a positive relationship, it is just as important to make sure you continue to focus on self care. Make sure that you are taking care of your own needs: physical, emotional, and spiritual, as this will benefit both you and them. You cannot help from an empty well and remember that the tool that you are looking to keep "sharp" is you.
Patience is another huge thing to remember when supporting your relative. Getting better takes time, even once the person is committed to treatment. Don't expect a quick recovery or a permanent cure. Be patient with the pace and be prepared for setbacks and challenges. Managing Bipolar is a lifelong process.
Accept both your loved one's and your limits. Remember that they cannot control their moods or snap out of a depression. Neither depression nor mania can be overcome through willpower, self control, or reasoning. Remember that each and every day they are doing their absolute best. Some days their best may look like them not getting out of bed, while other days they will seem like they are completely fine.
Know what you need to be able to lend support and that sometimes you may need to simply step away. That is ok as well. Remember that ultimately, recovery is in the hands of the person, not yours. You can be there to support them and cheer them on but you cannot force recovery (or even being seen by a professional) on them. These decisions are ones only they can make