E.,
This isn't a personality issue---it sounds like your daughter might have colick. Experiencing colic is probably one of the most nerve-wracking times parents can go through in their baby's development, but never fear: it does go away, usually within three months.
There are usually two reasons for colic: digestive issues and immature nervous development. In the case of digestive issues, this can actually very effectively be treated with a few visits to a pediatric chiropracter/craniosacral therapist. If this sounds a little "woo-woo", let me reassure you that scientific studies have been done in Europe that show a 70+% rate of resolving colic after a series of sessions. (My son has received craniosacral therapy since the age of two weeks and has been in good health; at 20 months, he's only been sick three times...but I digress.)
In regard to nervous development, because human babies are born with immature brains and nervous systems (or they would be too big for us to birth!), sometimes they have a tough time getting those systems "online". This can result in either a baby who is understimulated, who will then scream and cry to stimulate themselves enough, or a baby who is overstimulated, who will scream and cry because it's all a bit too much.
This can make for a horrid time for parents.
Some pediatricians will diagnose colic as GERD (gastro-esophagul reflux disease) and prescribe a medication for it. You can try that, but if you aren't seeing immediate results, this is not the correct treatment. At which point, a naturopath (who can help rule out allergens) or chiropracter would be other avenues to seek out for help.
I personally would try chiro first.
Some things for keeping your (and your partner's) sanity:
Take breaks whenever you can. Ask relatives with a loving pair of arms for breaks.
Buy earplugs. Lots of them. Leave them everywhere. You will still hear your daughter, but the noise will be muted. This is better for your nerves. (I've worked with several colicky kids, trust me on this!)
If you find yourself feeling angry or scared you may hurt the baby, leave the room. Go to a quieter place in the house and work off that anger. Scream into a pillow or punch it. Some days can be really difficult. Know what your limit is.
Keep track of yourself for signs of postpartum. A colicky baby can exacerbate the feelings of helplessness and rage. It's very important to ask for help and get it.
Check online about colic at the university websites. U of Michigan has a very good page of information. Just google it.
Ignore the endless comments from people who should know better but don't. Everyone feels welcome to comment on a crying baby, esp. in public, and tell you what you need to do. This can be hard, but it's important not to let strangers make you feel bad. There are a lot of people who assume that "quiet baby=good, therefore crying baby=bad". Try not to let the judgmental looks from strangers and family get you down. If some well meaning friend or family offers to "fix it"...let them try. It'll give you a break.
My very best to you-- I hope it isn't colic, but if it is, just two more thing to remember: IT WILL PASS and most importantly- This is not WHO your child is, it is a very hard space in their development they are passing through. This isn't indicative of their temperament: I've seen LOTS of colicky babies who cried 24/7 emerge into very sunny, fun kids. I hope this reassures you. Keep us posted!