Here's what you do! First look at the battery's sides, the ends, are they swelled out like a woman with wide hips? If they are, then the battery is bad or the alternator is over charging!
Put a charger on the battery over night. Make sure it gets a good charge! Then in the morning put the volt/ohm meter on the battery, what is the voltage? A good charge should be 13+ volts! If it's less than that the battery is bad! Next, start the vehicle, and with the volt/ohm meter still on the posts, see what the voltage drop is under a starting load! It shouldn't drop below 12 volts (not 100% sure any chilton/haynes might have this), if it drops significantly then the battery is bad!
Finally, with the vehicle running, see what the voltage is in the battery while the alt. is charging it! Should be 13.5 to 14 volts. Then turn on all the lights, heaters and anything else in the vehicle, and see what the voltage is at now! If the voltage is lower than 13 volts then the alt is bad! Generally the diodes in the alternator goes bad!
Now this is just a basic list of things I check for before I go and find a professional mechanic! I don't do the les schwab, schmuck's, or napa testers, I've had way too many of them tell me my batt/alt are good when they read false signals. If the battery is good, but not fully charged it will give the tester false readings!
One other thing to check for, is to look for a little sticker on the battery. It will have a letter and a number! The letter will represent the month of production, ie, A~jan, B~feb, and so on! The number will represent the last digit of the year of production, ie, 0~2000, 1~2001, and so on!
Hopefully this helps! I would lean towards a bad battery! If it only lasts five cranks, then I would assume that it isn't holding a charge! Which means the battery has been well used. Remember that heat is a batteries worst enemy! Heat from a motor, over charging, or drawing more from the battery than the alternator is replacing! All these will heat up a battery! And when the battery heats up, the lead plates will warp and start touching other plates causing shorts in the battery thus reducing the amount of voltage that the batt. will put out!