There is no simple answer to this, but a few words of advice: There is no contradiction between advocating for what you want and having a good relationship with the other person. Relationships are two-way streets, and in a healthy relationship, you should be able to ask for what you need. If you are so worried about damaging the relationship by approaching the other person, there's a good chance that the relationship isn't so healthy to begin with.
If you do decide to approach the other person, state what you want in terms of your underlying interests rather than by making a positional demand. Let's pretend that you find out that you are underpaid by 15% compared to others in your position. Instead of saying that you want a 15% raise, state that you would like to feel like the company values your work and your salary to reflect the market rate. Treat your boss as a partner in this discussion and ask for your boss's help in assessing what the market rate should be and how to go about adjusting your salary accordingly.
Throughout this negotiation, try to manage your emotions so as to remain friendly and calm, and spend twice as much time listening as you do talking. Treat the other person with respect, but not fear, and try to understand what is motivating the person's decisions. By understanding your boss's interests, you can help move the conversation in the direction you want it to go and make it more likely that you will get what you are looking for while building an even stronger relationship with the other party.