Bad Advice: Stop Trying To Be Friends with People
The first thing you can do is be kind to her at work, where you already encounter each other every day. Ask her how her day is going, and maybe invite her to join you for lunch if she tends to eat alone. If she does start struggling at her job, see if there are any little things you can do to help her out -- maybe offer to help her catch up on filing or boring administrative stuff everyone has to do, or if her job involves writing offer to help her by proofreading. Or, just walk up and ask if there's anything you can do, if she's looking really rough and overwhelmed one day.
From there, pay attention to her cues. Is she grateful and polite, but not forthcoming? If you give her an opportunity to open up (it's as simple as asking, "Is everything okay?") and she doesn't, drop it. If you invite her to spend her lunch break with you and she doesn't, drop it. If she refuses your help at work, drop it. Just the kindness you've shown her might have already started to lift her spirits even if she doesn't accept your help. You've checked off the box "Tried to Help Obviously Sad Person," and done a genuinely good deed. It's okay to stop there. If she either doesn't want or need your support, don't press it on her.
The thing is, maybe she does have a great support network. I know plenty of single people who live far away from their families who still have plenty of people rooting for them, and checking in on them, locally. Maybe she just prefers to keep her work and personal life very separate. Some people find comfort in that -- especially if she has a chronic illness, maybe work is the place she can come and not be asked about it all the time. Or maybe she is all alone but she just prefers it that way. Maybe she's all alone and doesn't prefer it that way but just doesn't want to be your friend for one reason or another. Whichever it is, just let her be.