OK, so here is what you are going to do. First, measure the battery voltage across the battery terminals, engine off, headlights off, everything off. You should see a voltage somewhere between 12.5 - 13.5 VDC, Now, while a helper turns the ignition key to the start position, measure the voltage (cranking voltage) across the battery. You should see a voltage somewhere in the range of 10 - 12 VDC whether or not the engine starts. If the engine starts, measure the battery voltage while the engine is running. It should be somewhere in the range of 13.5 - 15.2 volts. Next, place your DMM into AC volts mode and measure across the battery and see what the AC ripple voltage is. If the AC ripple voltage is under 30 mV AC, then according to most current reference sources, your alternator diodes are OK. If the AC ripple is in the range of 30 - 50 mV, your alternator diodes are questionable. Over 50 mV, then the diodes are likely defective depending on the manufacturer.
If the battery voltage goes below 10 VDC, you either have a bad battery, a defective alternator, a bad major electrical connection, or a bad starter. More on this in a bit.
If the battery voltage stays around what you measured when everything is off, then we are looking at a different problem that takes us down a different branch of the diagnostic tree. This branch will have us looking at the starter relay, the ignition switch, whether or not the ECM/PCM is supplying the ground signal to the starter relay control side, etc.
Back to the battery voltage measurements. Even a brand new battery can have a bad cell or an intermittent internal connection that can drive one to drink. Putting a big load on the battery will typically make this show up larger than life. If the battery voltage while running is below 13.5 VDC, the internal regulator could be defective. If the battery voltage while running is above 15.2 VDC, the internal regulator is toast.
If the battery voltage is normal while the engine is running then you'll need to measure the voltage at the starter while the engine is cranking (DMM negative test lead connected to the battery's negative battery terminal and the DMM's positive test lead connected to where the positive battery cable connects to the starter. This voltage should be the same or within a tenth of a volt or two of the battery voltage when the engine is cranking. If the cranking voltage at the starter is more than a few tenths of a volt different than the cranking voltage at the battery, then suspect a bad battery cable.
Before starting this make sure your battery is completely charged.
Please make the above mentioned voltage measurements and report back so we know which branch of the diagnostic tree we need to follow.
Good Luck!