Updating: try to no install a new kernel in an older computer.
do NOT use the utility included in sparky's UPGRADING KERNEL if the computer has 8, 10 or more years old or try to use the latest updating kernel in computer like this - (old) - why? because the new kernel give problems probably to the very old graphics card. Is more secure letting the kernel more old(posible) to the system; this can do update to all those programs using sinaptic or install "muon" (another handler of softwares) and have to the day all the system ( softwares) without risk of change the old kernel. This old kernel warranty your grapics card will run fine. IN other way YOU WILL need to reinstall the Complete OS again. Be carefull.
Note:I only did the first upgrading (recommended )when the initial installation. The kernel change to 4.8.0 working fine in KDE i686. I will not upgrade this kernel. For maintain stability. I will use only synaptic for software and general updating task as is recomended.
Another Tip: detail, never uninstall the old kernel if by need you has to install one more "up to date" or modern kernel . Uninstall the previous may cause that doing so takes "tangled" some files that is currently active - in other words leave the old installed. For not to run the risk of damaging something in the system.