Developmental pediatrician or pediatric neurologists can diagnose. I think also possibly a child psychiatrist? We went the developmental pediatrician route for our daughter.
How old is your child? If they are under 3, they can qualify for your state's Birth to 3 program, which is subsidized therapy. If they are over 3, they could qualify for developmental preschool... or if they are school aged, they would qualify for special help inside or outside of their regular classroom. I think the school system also makes their own diagnosis as well. Though it's better to get an outside professional's diagnosis. My daughter turned 3 in October. She was diagnosed with PDD several months ago, but the school system also diagnosed her with austism. The school system where we are didn't differentiate between the different types (aspergers, autism, or PDD-NOS).
I guess to give you more advice, we'd need to know more about how old your child is and what issues you have concerns about.
You would start with your GP or child's paediatrician and ask for a multi-disciplinary team evaluation of your child by professionals who have experience in diagnosing autistic spectrum disorders including Aspergers. This will usually involve your child being observed and assessed by a Speech and Language Therapist, Clinical Paediatrician, Clinical/Educational Psychologist, Occupational Therapist (if there are sensory or co-ordination problems).
You can also help by looking at the clinical diagnostic criteria to see what kinds of behaviours they are looking for. You can keep a record of any behaviours you suspect of being autistic in nature. You can write what your childs strengths and weaknesses are.
If you click on the Health Page icon on the top right hand corner of this forum page, go into The Behavioural Charteristics behind a Diagnosis of an Autistic Spectrum Disorder. I have posted the DSM IV Clinical Criteria and parents have posted examples of their child's behaviour that meets the criteria. This should give you some ideas of what the professionals are looking for.