the complete system w/ 6570 + celeron e3400 will use something like 150w load... as long as the PSU isn't crazy overrated it should be fine...
one of the reviewers states that the emachine uses a liteon 220w 80 plus bronze PSU
some other person w/ an emachine w/ 220w PSU and Athlon IIx3 ran a 5750 off theirs
http://www.tomshardware.com/forum/296987-28-upgrade-dual-core
i'm assuming that person did not die in a horrible explosion, and that it worked fine and he never reported back
so the OP should be fine upgrading to a 6570
also like notty suggested, i wouldn't recommend upgrading the CPU, the only upgrade worth it would be a core 2 quad, and it'd cost a lot, wouldn't be worth the hassle (likely buying used) (remove hsf, remove CPU, place CPU, thermal paste, place HSF), and would likely bring you over or at least close to the wattage limit