Undo the battery cables. Remove the battery. Put the battery on the battery charger till it has a full charge. Let's make sure you have a battery that will hold a charge. New batteries can be bad and if you have lowered the specific gravity in it enough by draining it a charge may not occur. While it is charging you can look at the charging system of the lawnmower.
Follow the negative cable to where it is bolted to the chassis. Remove the bolt holding the negative cable to the chassis. Sand the area around the bolt hole with until shiny metal is visible, put a thin layer of oil on the metal and the end of the cable. Attach the cable back to the chassis using a new bolt and lock washer. Make sure that connection to the chassis is tight and you can't move the wire.
On the starter solenoid one of the connectors has 2 red wires. The thicker one runs to the battery. The thinner one runs to the ignition switch. Make sure the connection there on the solenoid for those 2 wires is clean and tightened properly. Now follow the thinner wire to the ignition switch, before you get to the ignition switch you will find a 15 amp fuse, make sure the fuse and connections there are good. Make sure that wire has good connection to the ignition switch. There is another red wire on the connection where the previous one is attached. Follow it to the magneto. Including the one you are following the magneto has 3 wires coming from it. Make sure the connection there is good. Reattach the battery making sure the wires are firmly attached and you can't move a wire. Start it up run it for at least a half an hour.
If the battery was able to take a charge and it doesn't charge now either you didn't tighten a wire down enough or the magneto is bad.
From the ignition switch, follow one of the red wires to a fuse holder. Check that fuse and replace it if necessary.
Check the other red wire and follow it to a 3-pin connector. Make sure that connector is fine, and then keep following the wire to a diode. If everything else is good, then you might try replacing that diode. It might be internal to the motor, though.
I found the parts diagrams for your mower by searching for murray + the model number. They showed the wiring diagram. You can also get replacement parts there.