Slaves did, on occasion, join the army. But as u/TheMuffinsPie said: this was illegal. To quote one of Pliny the Younger's letters to emperor Trajan:

THE VERY excellent young man Sempronius Cælianus, having discovered two slaves among the recruits, has sent them to me. But I deferred passing sentence till I had consulted you, the restorer and upholder of military discipline, concerning the punishment proper to be inflicted upon them. My principal doubt is this, whether although they have taken the military oath, they are yet entered into any particular legion. I request you, therefore, Sir, to inform me what course I should pursue in this affair, especially as it concerns example.

Trajan's answer is quite clear on what ought to be done:

SEMPRONIUS CAELIANUS has acted agreeably to my orders, in sending such persons to be tried before you as appear to deserve capital punishment. It is material, however, in the case in question, to enquire whether these slaves enlisted themselves voluntarily, or were chosen by the officers, or presented as substitutes for others. If they were chosen, the officer is guilty; if they are substitutes, the blame rests with those who deputed them; but if, conscious of the legal inabilities of their station, they presented themselves voluntarily, the punishment must fall upon their own heads.

Still, it wasn't as clear-cut as all that. In desperate times, the Romans would enlist slaves in their legions. These would either be freed immediately, or promised their freedom if they served faithfully. Roman doctrine, in general, held that only free men could be expected to fight well. (This also went for the navy, which is why the whole "galley slaves" concept wasn't a thing in the Roman fleet.)

For example, Augustus raised several units of auxiliary troops from manumitted slaves in the crisis years of 6 and 9 A.D. Although freed slaves automatically attained citizenship, Augustus nonetheless didn't want these men to serve in the legions proper. The auxiliary cohorts they served in were named Cohortes Civium Romanorum or "Roman Citizen Cohorts" and retained something of a special status even after the original crop of slave-recruits had long been phased out.^1

More famously, after the disastrous Roman defeat at Cannae, the senate appealed to household slaves to join the legions in return for freedom. 8000 of these volones joined up and were formed in two legions. Of course, this being mid-Republican times, these legions were later disbanded as usual, so there was no long-term unit heritage like with the Augustan cohorts.^2 (Incidentally, Livy also tells of the Romans enlisting 6000 criminals and debtors under similar conditions, and arming them with captured Gallic weapons and temple trophies, among other things. Desperate times...)

There are several more examples to be found, but all of these are exceptions to the rule. In general, no slaves could join the legions and use them for social mobility.

However, there were many slaves in the army. In Imperial times, we know that there were officers and enlisted men sometimes had personal slaves, but there were also official army slaves, with tasks like taking care of the legion's animals and baggage, and so on. These slaves even seemed to have had some training and equipment and sometimes fought! See my post here for more details.

Finally, even though joining the army wasn't really an effective means of climbing the social ranks for slaves, manumitted slaves in general could in some cases reach quite lofty positions. Indeed, many early emperors such as Claudius relied heavily upon their freedmen to administer the empire and run its budding bureaucracy.

• 1 Goldsworthy, Complete Roman Army p.64
• 2 Goldsworthy, The Fall of Carthage p. 219

Although I'm not an actual historian, I feel like I have enough knowledge on this subject to answer this question. However, if any actual historians could fact check me, or expand upon my answer, that would be great.

Roman slaves would not be able to become land owners through the army because they could not join the army while they were slaves. Slaves were the property of their owners, and we're not eligible to join the army. They were unlikely to be able to run away and join the army as well. Slaves that were caught running away from their owners were typically branded with the letters FUG on their foreheads, short for the Latin word fugere, meaning to flee.
However, slaves could join the army if they stopped being slaves. Masters had the legal ability in Roman society to free their slaves, and let them become fully fledged Roman citizens. From there, if a former slave wanted to join the army and serve until they were rewarded their land, it was entirely possible.

Thanks a lot for the answers !

Provided that you were a free man, (former slave or not), could the army allow you to become a respected citizen and, let's say, your son or grandson to involve in politics ?