It's possible that she is just being dramatic and doesn't actually intend to hurt herself. If so, great, but there's likely still some truth to what she said: she feels like she isn't getting attention, and/or she feels like she isn't valued and loved. The fact that she's been increasingly sensitive to criticism indicates this if nothing else.
However, it's also possible that she is truly depressed. Young children do attempt (and succeed at) suicide (ref. Suicidal Behavior in Children Younger than Twelve: A Diagnostic Challenge for Emergency Department Personnel) So be glad that she is reaching out for help, because it's a chance to figure out what's going on and intervene.
Things I would do as soon as possible:
Consult a professional. Don't dismiss this as "just a phase." A child psychologist knows the right questions to ask and what to look for. Regular appointments with a therapist can provide her a chance to talk with a non-critical adult (she can complain about your or her mother, for example, which she may not want to do to you directly) and a professional can teach coping techniques to help her deal with the root causes of low self worth.
Talk to her. Be honest about why you want to have this conversation ("Child's mom heard that you told Child you're unhappy and you don't like yourself"). Offer her the chance to open up. Focus on your concern for her, and avoid any suggestion that she's done something wrong by sharing her worries, fears, and/or depression.
- I'm not angry. I'm just worried and it makes me sad to hear that you're sad.
- I'm glad you told somebody that you feel unhappy because now I can try to help you.
- I want you to know that you can talk to me about things that make you unhappy.
Ask her what brings on those feelings of low self-worth. Be prepared to hear that it's something you do (my son gets intensely miserable and self-hating when he feels neglected), and be open to changing yourself. This doesn't mean you're a bad parent, or that this is all your fault (or all her mother's fault, or all whoever's fault), and blaming is counter-productive.
Include other adults. Her mother needs to know about this. (Depending on your relationship, it could be a tricky conversation -- just avoid blaming and focus on your daughter -- but advice on that is outside the scope of my experience and this Answer.) Depending on how severe underlying issues are, teachers and school counselors may also need to be brought in (especially if there's a bullying or she's overwhelmed by schoolwork).
Some general ideas for how to increase her sense of self-worth on an ongoing basis:
Focus on positives. Praise effort, not "her" (for example, "Great job on your homework, you clearly put in a lot of effort" instead of "Your homework is correct, you're so smart"). Thank her for doing chores or little helpful things around the house ("It's so helpful when you [set the table, tidy your room, fold your laundry]"), even for things that she is expected to do. Show your appreciation for the difference she makes in your life.
Remind her she is loved. Hugs, praise, and including her in your life, both in special and everyday ways. Taking her out for a daddy-daughter dinner date is a great occasional treat, but everyday things are just as important.
- Show up for practices or performances (sports, music, dance, whatever).
- Sit with her while she does homework instead of sitting her somewhere and expecting it to happen.
Ask for her help with miscellaneous tasks you're doing. Find age appropriate ways she can help, even if the overall task is not something you'd want a 6 year old to attempt solo.
- Want to do the dishes with me? (Bubbles are fun.)
- Want to cook [favorite meal] with me? (SO MANY ways to help in the kitchen!)
- Want to help me write this email? (Let her type words.)
- Want to help me fix this light bulb? (Talk about electrical safety, talk about how light bulbs work, ask her to hold the new bulb while you take the old one out, etc.)
Listen to her. Pay attention to her interests.
- Let her take the lead on suggesting activities (going to the park, playing with dolls, cooking, hammering random pieces of wood together). Keep an eye on what she wants to do, and try to figure out ways to make that happen more often
- Let her ramble on about a book or movie or toy that she's excited about. Nod and smile and make interested comments: "Wow," "That IS cool," etc. (Be prepared to be bored. Keep listening even when you are bored.)
Use constructive criticism. Kids need guidance and teaching, and we don't expect them to know everything. Unfortunately, some kids take this badly and as evidence that they Just Can't Do Anything Right. Focus on things she's doing right, concentrate on the goal, praise her effort and eventual success. Ask leading questions "Hmm, is that the right way to do it?" to help her examine the problem herself and self-correct, rather than you doing all the critiquing and correction. There are a couple other Questions that may be useful to look at: