2001 Mitzubishi Spyder Eclipse ONLY STARTS WHEN JUMPED!

Two friends have said two seperate possible reasons why…1) the alternator may be dead–need a new one, and 2) the computer system has a glitch in it. When jumped, the car starts (barely), and after running for a while (an hour at least), it still does not maintain enough power to start again thereafter. New battery or new alternator? Know of a way to check these things out myself?