Well, he most likely didn't. There's no proof Tiberius killed Germanicus, to the point - contrary to common belief - not even hostile authors accuse the princeps of such misdeed. Suetonius, for one, never openly accuses Tiberius of that, but tries to be more cautious:

It is even believed that he (Tiberius) caused his (Germanicus') death at the hands of Gnaeus Piso, governor of Syria

- Suetonius, Tiberius, 52.3

Then chosen consul for a second time, before he entered on his term he was hurried off to restore order in the Orient, and after vanquishing the king of Armenia and reducing Cappadocia to the form of a province, died of a lingering illness at Antioch, in the thirty-fourth year of his age. There was some suspicion that he was poisoned; for besides the dark spots which appeared all over his body and the froth which flowed from his mouth, after he had been reduced to ashes his heart was found entire among his bones; and it is supposed to be a characteristic of that organ that when steeped in poison it cannot be destroyed by fire.
Now the belief was that he met his death through the wiles of Tiberius, aided and abetted by Gnaeus Piso.​

- Suetonius, Gaius, 1.2-2.1

Tacitus is even less explicit. He never states that Tiberius poisoned Germanicus, but rather reports sinister rumors. More importantly, the whole narration of Piso' alleged mission to check Germanicus is built around insinuations, and is not treated as a fact.

But, apart from the paternal temper, Piso's brain was fired by the lineage and wealth of his wife Plancina:​ to Tiberius he accorded a grudging precedence; upon his children he looked down as far beneath him. Nor did he entertain a doubt that he had been selected for the governorship of Syria in order to repress the ambitions of Germanicus.

- Tacitus, II.43

The same reason brought forward by Suetonius to suggest a murder is pretty much rejected, and the whole poisoning theory is treated with scepticism.

Whether it bore marks of poisoning was disputable: for the indications were variously read, as pity and preconceived suspicion swayed the spectator to the side of Germanicus, or his predilections to that of Piso.

- Tacitus, II.73

The single charge which he seemed to have dissipated was that of poisoning. It was, indeed, none too plausibly sustained by the accusers, who argued that, at a dinner given by Germanicus, Piso (who was seated above him)​ introduced the dose into his food. Certainly, it seemed folly to assume that he could have ventured the act among strange servants, under the eyes of so many bystanders, and in the presence of the victim himself: also, he offered his own slaves for torture, and insisted on its application to the attendants at the meal.

- Tacitus, III.14

Tacitus' final judgement is as equilibrate as it gets.

This closed the punitive measures demanded by Germanicus' death: an affair which, not only to the generation which witnessed it, but in the succeeding years, was a battle-ground of opposing rumours. So true it is that the great event is an obscure event: one school admits all hearsay evidence, whatever its character, as indisputable; another perverts the truth into its contrary; and, in each case, posterity magnifies the error.

- Tacitus, III.19

Going beyond rumors not even ancient writers reported as facts, we have little reason to believe Tiberius poisoned Germanicus. The whole latter's career perfectly fits within the standard of imperial heir, including Tiberius himself. The Augustan precedent is pretty clear, and Germanicus had proved himself to be loyal to Tiberius. We don't know whether Tiberius really disliked or mistrusthed Germanicus - it's possible he had a low opinion of his campaigns in Germania, and disliked his impulsive behaviour, but whatever his sentiments were, they never reflected upon Germanicus' very career. On the other hand, Tiberius had little reason to trusth Piso: As ancient sources claim, he was no friend of Tiberius, nor he was a friend of the Domus Augusta. Why he should have trusthed such a man with poisoning his loyal adoptive son and nephew goes beyond our immagination. Apart from sinister, unsustained rumors there's nothing that suggests Tiberius poisoned Germanicus, and the idea is pretty much rejected among modern historians. Germanicus's death proved to be tragic and untimely, but most likely natural - and Tiberius was certainly not pleased by it, even solely for the fact it destabilized the future of the Domus Augusta. On the other hand, he honoured the memory of his deceased step son in lavish ways, as epigraphic evidences such as Tabula Siarensis proved. In short, we're contemplating the career of a successful Caesar, cut short by some illness, of even by a careless physician.

As for your question, the Netflix documentary is absolutely terrible. You should watch it as some kind of fiction. This 'review' takles all the major flaws in the first minutes of the Gaius series: https://tl.net/blogs/556411-a-historians-rant-about-netflix-roman-empire

Imperial succession in ancient Rome did not work as an hereditary monarchy, especially in the early principate. Rather, a princeps, the pater familias of the Domus Augusta, trained a 'pool' of youths as potential heirs by slowly involving them within State business. As a matter of fact, Drusus, Tiberius' son, was already contemplated within 'succession' plans, was slowly gaining experience on par with Germanicus and it's likely he was intended to succeed Tiberius along with Germanicus himself. Julio-Claudians tried to put forward a double succession more than once. Augustus, for one, had Tiberius himself and Drusus the Elder, then Gaius and Lucius, then Tiberius again and Posthumus. Tiberius will try to have Germanicus and Drusus, Nero and Drusus Caesar and Gaius and Gemellus. Claudius, on the other hand, will try to put forward Nero and Britannicus. All of them, of course, will sorely fail.