Share the job
You may have got into the habit of believing "it's your job and no-one else's" to support the person you care for. Look at how you can ask for support from others. This applies not just to the practical aspects of caring, but also to the "job" of providing emotional support. You may find it helpful to suggest to the person you are supporting that they talk to someone else as well as to you - such as a counsellor - about their feelings. This may actually be better for both of you.
Don't try to "solve" the unsolvable
Some carers feel it's up to them to find a solution when the person they are supporting is unhappy, angry or upset about their illness or its treatment. Actually, they may prefer it if you don't try to provide answers but just listen and understand so they don't feel so alone.
Don't overprotect the person you are supporting
You can actually do too much for someone. When this happens, not only are you likely to over-extend yourself, but the other person can feel powerless and dependent. This can leave them feeling even more frustrated with their illness.
Watch your stress levels
Taking regular exercise and learning relaxation or meditation can help to reduce stress and make you feel better able to cope with being a carer. You'll find more about stress in The emotional effects of kidney failure.