"Sitting a few years". A lot can deteriorate in that amount of time. I don't think buying a battery, checking a few minor things and trying to crank it stands a chance of success on a tractor that has stood in the briar patch for years.
Seems like someone here would offer some methodical suggestions on how to attack the problem. Unfortunately I'm no mechanic, so I can only make some random comments. Perhaps my inadequate reply will stir someone competent to respond to you. C'mon Guys! You LOVE to give advice!
I think I would want to try to make sure the most difficult/expensive things to fix are checked first; besides, if it DID fire, you might ruin other stuff - unlubed bearings, electrical system (shorts from exposed wires), etc.
Face the fact that it it is going to take an investment of time and money to get this machine running, (how much, I can't say - not a lot if you are lucky and methodical) and you may get a great tractor for you efforts! There is no way this machine is going to run without some effort. Even if all the running gear is functional, you must have an working fuel and air intake system, and an electrical system that will warm the glow plugs and at least turn over the starter.
Google " turn diesel over by hand" and find out if the engine is locked up. Some good tips in some of the hits. Probably moisture in the cylinders has caused rust that makes them difficult or impossible to move.Then do similar searches for other parts of the system. Maybe you can find a methodical checklist for evaluating a neglected diesel.
Diesel (?) in the radiator suggests a bad cylinder head gasket or cracked block to me. Soot in the air cleaner suggests a severe blowback (engine timing problem?) that could have damaged other things (happened to a gas engine I had). I'd want to get some idea of the causes before even thinking about starting. Why did the previous owner walk away from it, apparently without trying to fix it? Does he know something he has not told you?
The fuel system is bound to have gelled diesel in it, as I bet it was not drained. Heck, that may be why the previous owner could not start it (did he know anything about tractors?) At some point, replace all the fuel lines, clean the carburetor thoroughly, new fuel filter, clean tank and more.
You did not mention if the starter solenoid and starter work (short the battery + to the solenoid to test) after cleaning dirt out of the alternator. Is the wiring insulation brittle throughout (potential shorts)? Expect bad grounds.
Make a list of what MUST be carefully checked, and another of what MUST be serviced/fixed, before attempting to start the tractor.. Then check module-by-module, ensuring there are no problems, or that they have been corrected. Then you wil know your tractor inside-out, and will be able to run it with more confidence than most of us.
Good luck!
C