A behavior chart is a good idea and it's very good that she came up with it herself, so she feels ownership of that idea. I do believe that parents have to be responsible for their own behaviors toward their children and should apologize to their kids when it is called for; however, take care that her idea of making the chart for both of you isn't an indication that she would consider her own unacceptable behavior somehow OK if you don't do what she considers your end of the bargain.
I'm concerned that at the age of 7 she has so little anger management and is hitting. A few questions only you can answer:
Does she do this at school? Have you talked with her teachers (all of them) and the school counselor to see if this is going on there? Does it happen on play dates? It's possible she may have had outbursts at school that you haven't heard about because they didn't involve her hitting another child or a teacher, but angry outbursts even without the hitting should be something you ask teachers to tell you about the same day they happen. That way you can track how often and in what settings this goes on. You will have to invest some serious time in talking with the teachers and counselor and you should do it in person even if you have to get a babysittter for the twins -- the teachers may not realize how serious you consider this if you just shoot them a quick e-mail. You will need to get the counselor involved and the counselor should be glad to see you alone, without your daughter, to talk about this and offer some help.
Does she seem jealous of her twin siblings? Does she ever threaten them with hitting, or even just verbally vent her anger at them? Having a little sibling is enough reason for natural jealousy, but having twin siblings may seem to her like she'll never get anyone's full attention again.
What sets her off? Not getting what she wants immediately? If so, possibly she has grown used to getting things when she has tantrums -- in short, tantrums have worked well for her all her short life. Or is she more prone to the violent tantrums when she's tired? Stressed? Hungry? All three? Is she over-scheduled with activities and therefore worn out? Have the tantrums increased since she started school, which might indicate she's stressed by that somehow? Have there been life changes recently -- are the twins new babies, have you moved, etc.?
Are you the only person towards whom she has these outbursts, or the only one she hits? You didn't mention whether you're married -- If so, how does she interact with your husband or significant other? Would she try these tantrums with him? Does he act extremely strict with her, so she's scared to fuss at him but not at you, or is he extremely indulgent, explaining away her outbursts as "She'll grow out of it" or "Well, she's mad, let her vent" etc.? If you and he (or you and Grandma or any other adults she sees regularly) are not on exactly the same page about how to deal with her, you won't be effective.
It's excellent that you and she sat down and talked about this. Praise her a LOT for talking with you and initiate talks with her about her feelings often. I recommend the American Girl "Feelings Book" for you; she may be too young for it yet and it does contain issues such as feelings about suicide, etc., that you may not want to discuss with one so young, but it does have great ideas about anger, frustration, depression etc. that will help you talk to her. It may be that she cannot express feelings any way other than tantrums yet and it's time for her to learn to work through things better. You can help with that and so can a counselor, either private or at the school.
When she starts getting angry, try a method to slow her down before it escalates, and work out that method with her beforehand, when she's calm -- maybe you can arrange a "signal" between you like your saying "Let's both take a deep breath" as her cue to be quiet and think before going on, etc. I know there are books out there that have lots of methods to halt tantrums BEFORE they escalate.
I'm not sure how you discipline her, but with some kids (including mine) it's hard to find the "currency" that makes them change behaviors that are so ingrained or that make them feel temporarily in control. (And your daughter may do this because it makes her feel strong when she usually feels like she has no control over anything.)You'll want to find effective discipline that really means something to her, like taking away certain things (TV time, computer time, play dates, etc.) that she values highly. And giving her more responsibilities, then praising her a lot (even for what seems like small stuff to grown-ups) could help turn her around -- she may indeed feel like she has no say in her own life. Just be sure the responsibilities are realistic and doable for her.
One other thing: If you or your husband spanks her, consider that, to her, it sends the signal that physical reactions are acceptable to adults--so why shouldn't she resort to getting physical when she's upset? Even if you tell her that it's a discipline for her behavior, and you expect it should correct the behavior, in HER young mind it says "it's OK to strike someone."
You're doing great with the calm talk and the chart, but consider following up with looking at when and why she seems to have most of the tantrums, talking to a professional if they go on much longer, and getting her teachers involved in informing you about the rest of her day.