Leckhampton and Charlton Kings are both expensive. Of the two, CK is a better bet school-wise, because you can choose a location that gives you a really good chance of getting into a good primary (Glenfall) or infant/junior (CK) school followed by the secondary (Balcarras). Leckhampton is trickier - there are three oversubscribed primaries (two are church, but with no faith admissions criteria), so you need to be fairly close to get into one of those. But those areas don't overlap much with where you need to be to get into Bourneside, and I'm not sure they overlap at all with Balcarras's effective catchment.
Over subscription depends very much on area and school. Two of the secondaries for example are always oversubscribed, two are not, and the fifth is Pates, which doesn't really count.
Both have good community feel, although it depends a little where you are. I used to live at the village end of Leckhampton rather than Bath Road, and it felt like friends at the Bath Road enjoyed more community spirit - possibly cos I'm a bit antisocial though. Or maybe because most of the pubs are up that end! CK is weirdly isolated from the rest of Cheltenham, very villagey, mostly in a good way.
Flooding ... The A46 running between Shurdington and Leckhampton flooded in 2007, which affected access. There was little in the way of residential flooding - some houses on Church Road, which was to do with the individual houses being in a dip rather than wider flooding. CK has the Lilley Brook running through it, but it's mostly very deep-set compared to housing. Our house came up in searches as within 100m of a waterway but with pretty much zero chance of flooding because the brook is so far down compared to houses.
Cleeve I don't know well at all. It's a lot cheaper, the schools are good, the secondary seems to have done away (for now) with very complex fair banding admissions criteria. Everyone I know who lives there likes it. In part it depends on whether you want to be in Cheltenham proper (the other two areas are walkable/cycleable into town) or if you don't mind being in more of a satellite (in which case, depending on the work reasons, you might want to consider parts of Gloucester).