you could also consider microdialysis or fast scan cyclic voltametry (FSCV) which would allow you not only to measure concentrations but dynamic changes also, however with FSCV you will only be able to detect some neurotransmitters, including dopamine. If your institution have the adequate equipment I would also recommend 1H MRS that can be used to monitor a number of neurotransmitters (global concentration) including glutamate & GABA (but not dopamine) from 1 or two brain regions simultaneously.
Indeed, HPLC-ECD is a very common analytical technique for measuring total neurotransmitters in brain regions. Luis has given a very detail protocol for the assay. I would also suggest to add internal standard (IS) in the PCA (with cysteine) for the protein precipitation step. The applied cell voltage should be around +750 mV.
If you do not plan to do it routinely, you can just send your samples to some facilities that are specialized in HPLC measurement. We did this for our olfactory bulb biogenic amines (including dopamine, 5-HT, adrenaline, noradrenaline etc.) measurement at Vanderbilt University VBI/VKC Neurochemistry Core. All you need to do is to dissect out your samples, fast frozen them, then overnight ship to the facility.
Luis gave are fine detailed answer! Supplementing: We used HPLC-MS/MS for about 15 different neurotransmitters from different brain regions with ESI pos or neg measuring to extremely low levels. HPLC-MS/MS has a much faster stop and go time as HPLC-ECD!