You don't say which battery cable. Is it the positive from the battery to the starter solenoid? If the cable ends are hot it is probably a bad connection/corrosion there. If the cable itself is getting hot along its length, you have excessive amp draw. First recheck all wires you touched when doing alternator. Are they in right place according to the book? Always suspect your recent work when a problem occurs afterward.
Voltmeter tests you can do.
1. Put voltmeter leads on battery positive and negative when battery is at rest. Set at 20 volts. Should read over 12.5 v. ignition off, no draw. Turn on key and headlights. Voltage should not drop more than .4 volts. if there is a big drop, battery has a bad cell.
2. Visually check all connections and starter to frame ground to make sure contacts are all good.
3.Turn key to start, should be over 9.6 v. If below that battery has a bad cell or excessive starter draw.
4.Test battery with load tester before going further. A weak battery producing low voltage will cause excessive current to flow to starter and heat up wire.
Battery must be good for rest of tests.
5. Put one voltmeter lead on battery positive and other on end of positive cable where it attaches to starter solenoid. Leave meter set at 20v. range
Crank starter. If voltage is over .2 volts connections are bad. Over 1.0 volt is a bad cable. (You are measuring voltage drop in cable under load.)
6. If Test #5 shows less than .2 volts , leave one voltmeter lead on positive connection at starter solenoid. Put other lead on negative terminal of battery.
Meter should read within .1 volt of battery voltage. Crank starter. If voltage drops to 9.6 volt or less, and battery is good, starter is drawing excessive current.
Take apart starter, make sure lubrication of motor shaft is good, and armature is not rubbing on case or windings somewhere and brushes are free and clean off dirt on brush slides.
If the voltage stays above 11.5 volts or more and starter is slow or won't turn, there is an internal problem in starter.
If no visible problems, take starter to rebuilder.
7. To test ground circuit. Put one voltmeter lead on some metal part of the starter. make sure you have a clean connection. Put other voltmeter lead on battery negative post. meter in 20 volt range. Crank starter. If voltage is over .2 volt you have a bad ground connection somewhere. Make sure starter bolts to engine are clean and tight. Check and clean connections on ground strap from engine to frame. Check and clean end of negative cable where it bolts to frame. Check negative terminal on battery.
Probably you have something wrong related to your recent work or battery has a bad cell, but these voltage drop tests will let you isolate a problem quickly. If you replaced the alternator, I assume the battery was getting low and repeated starting with a low battery draws heavy current and can damage starter or cables.