Firstly are you absolutely certain that the LA social services are seeking a permanent home for this child. Not all children in short term foster care will need permanent homes, as some will return to their own families. If the child does need a permanent home the LA have a duty to find out if there is anyone in the extended family who could care for the child on a permanent basis. Dependent upon the age of the child and the particular behaviour problems she has, even if an adoptive family is not found, there is no guarantee that a permanent foster family will be found.
I don't know the age of the child, nor the age of your own children (if you have any) but as I'm sure you know, this is not a decision that you can make lightly. It would be very different to have a challenging child 24/7 than seeing her for a few hours in the school day. There are many children who are awaiting permanent homes, and I wonder if you have thought of adopting before, or whether the thought has only arisen because of this child.
You would need to contact the LA and speak with the social worker who has case responsibility for the child. Much will depend on your own circumstances and you will need to attend an Adoption preparation course (usually run for one evening a week for about 6 weeks) if the LA think that you are someone who could possibly give this child a permanent home. After than if you still want to go ahead, you will have to undergo a comprehensive "homestudy" where an adoption social worker would look at your own childhoods, your parenting capacity (if you have children) your marriage/partnership, lifestyle, employment etc. When that is complete your application is presented to the LA Adoption Panel (which is a multi disciplinary panel) and if approved, you then wait for a suitable "match."
It is unusual for applicants to be considered for a specific child but not beyond the realms of possibility. I don't think the LA would be in a position to let you know the details of the child's background and reasons for being in the care of the LA until after you have been approved by the Adoption Panel, as they are bound by confidentiality.
I would urge you to really consider this "offer" - all children awaiting adoption are going to be to a greater or lesser extent challenging because of their pre placement experiences and many people first considering adoption think that if you love the child and give them a good home all will be well, but sadly this is often not the case. These children have learned that adults are not to be trusted and so will not necessarily settle into a new family. Adopters need to understand that dependent upon the extent of the abuse or neglect that the child has suffered, they are in for a "long haul" of being tolerant and patient and resilient in caring for these emotionally damaged children.
All of this will be discussed at the preparation group. I'm not sure if you are a teacher or a TA. I don't think the issue of the child being in the same school is a problem but obviously if you do adopt this child, you could not also be her teacher or TA.
Hope I haven't put you off, but I'm just being realistic.