How do I build a relationship with a professional?

Two big things.
First, commit time to the relationship. Usually, this means working as either a volunteer or intern.
Second, focus the conversation more on the other person than yourself. For example, instead of asking "How should I...?" Ask "How did you ...?" Even better if you can say something about their story before you ask another question.

Consider how you are reaching out to professionals and try refining your approach differently until you find an approach that works for you. When I began my job search, it took many tries to get the "informational interview" email text just right. If you are talking in person, have a script ready that focuses on the professional: you want to learn about their experiences (humbly), and are eager to learn more about their typical day, and what enticed them to this career.

Also consider how you are meeting these professionals, and if there is a different setting to meet professionals that will yield better results. Are there organizations in your field that you can join and meet professionals that way? Does your school offer any opportunities where professionals with a more mentoring mind-set may participate?

You may not be trying this, but I caution against asking for letters of recommendation from anyone unless you have worked for them. You may inquire if they know of opportunities for internships or jobs to get more experience and test whether it is a good fit for you (which will lead to letters of recommendation from an employer). Best of luck!

Develop Your People Skills. Good relationships start with good people skills. ...
Identify Your Relationship Needs. Look at your own relationship needs. ...
Schedule Time to Build Relationships. ...
Focus on Your EI. ...
Appreciate Others. ...
Be Positive. ...
Manage Your Boundaries. ...
Avoid Gossiping.

Hello Jackie:
This is a good start...you are seeking help!
This means that you have been trying and you are open to suggestions. Maybe doing it on your own is not much help, thus-you seek advice from others.

First thing first, try to look at yourself in the mirror. Smile, talk, grump, laugh. Now, ask yourself, "do I look pleasant?" or "do I look unapproachable?" and be honest!
If you feel that your facial expression or your body language already scare people away, then that's the giveaway! You may not be getting good vibes (vibration) from people around you because your demeanor and the way you talk may not be appropriate. On the contrary, if you see yourself pretty, charming, confident, and soft spoken, perhaps, then you are on your A game.
As per advice from Mr. Shetty and Ms. Reuter, work out on your people skill and interpersonal skills. Socialize, learn from good leads and drop bad leads. But dont forget, keep you as yourself! Same birds fly together!

Letters of recommendation can be requested from places that you worked (so long you performed well), from volunteer experience, and from your professors and your student advisor. So long as you had shown respect, professionalism, and good work ethics, I don't see why they should not give you.

Best of luck!
Maria