The Opti 780 has 95W CPU limit. Generally the best CPUs are the 1333fsb 45nm quads.
Here are the builds for those.
http://www.userbenchmark.com/System/Dell-OptiPlex-380/1942
It helps to know what size the Opti 780 is. thay came in 4 sizes. The best is the Mini Tower MT, then the Desktop DT, than the Small form factor SFF, and the worst is the USFF.
The 240W PSU makes me think it's not an MT, but a DT or SFF. The 240W PSU willl power a 95W CPU and up to a GTX1050Ti if you canfind one that fits.
You most likely will need a cooler upgrade to run the quad. This varies with the size case. The 2 cores came with an Aluminum lump for a cooler.
There are acouple of fun 2 core mods for those, and a q6600 overclock but the best setup is a Q9650, or Q9550 CPU.