Choosing the right bullet, picking your shots and hitting where you shoot at are important with most any caliber. It's no less true for the .243. My first CF rifle was a Ruger M77 in 6mm Rem, back in the mid '70s. It's pretty much a ballistic twin to the .243. Still have it, and it still does just fine, it's taken several whitetails cleanly with 100 grain bullets.
DW got her Winchester Model 70 Youth model in .243 as one of her wedding presents from me, and the next season (we got married in Dec) she collected her first deer with it. Not ideal shot placement, she hit too far back (later she said something about too many people insisting she shoot center of mass for too long) but the little spike only went a few yards before bleeding out. She swears she'll do better with placement next time, but so far there hasn't been a 'next time.' She loves shooting the little rifle, too, and that doesn't hurt anything.
Best answer I can give is, pick good bullets, and from there on, it will do if you will do.
lpl