Here are the factors you need to consider
DCPS has PK3 and 4 but they are by lottery and you are not guaranteed a slot at a neighborhood school. The most desirable schools are hard to get into and you most likely will not know until August if you have a lottery number over 20. You must apply through the lottery process in late January
Charter schools are publicly funded but allowed to run their own admission process, for desirable schools usually a lottery. Similar issues on timing to the desirable DCPS schools. Call the schools to get the information and confirm on website.
You do not say where you are from, but many folks have a bit a sticker shock when the see the price for private child care. So the public options can be a life saver.
Consider your commute carefully, there are some pretty awful bottlenecks and may or may not be before care.
Kindergarten into a desirable school may also require a lottery if you don't get in at the PK level. For the most desirable schools there just won't be slots because sibling preference will take that slot. Ask the school how many they accepted last year to get a sense.
Test scores are often a proxy for SES and the rich poor gap in this city is extreme. That said it unfortunately affects opportunities for enrichment versus remediation in most schools. However when you look at what most of us are saying about schools have a salt shaker, we are a neurotic bunch and anonymity can make some of us frankly sound like wackos. Ultimately a lot depends on what your child will need and their personality.
These websites can also be helpful.
http://downtowndckids.org/charters-which-are-the-best
http://downtowndckids.org/charter-schools-introduction
http://downtowndckids.org/introduction-to-dcps-the-process