Tibby is one of my favourites
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It depends what you necessarily want to know, so PM me for more details.
But a basic run down:
He was the fourth choice I think, and was reluctant to hold the position of Princeps, and refused many of the titles that Augustus had (such as Pater Patriae or Father of the Country) apparently so that he wouldn't be compared to Augustus, and to avoid offending the Senate.
Sejanus- Prefect of the Praetorian Guard, referred to by Tiberius as the "Partner of My Labours". Well trusted by Tiberius, but actually plotted behind his back, killing people off etc to try and get to the throne. When Tiberius finally found out, he was devastated and this caused the Treason (Maiestas) trials, in which he became suspicious of everyone. Increased role of the delatores (informers).
Capri- Sejanus saved Tiberius from falling rocks, which solidified Tiberius' trust in him at that point. Sejanus also had a lot of control as people could only see Tib if they went through him first, and even when Sejanus was killed, Tib stayed in Capri.
Praetorian Guard- Regrouped into one large cohort rather than nine (?) and brought within the empire rather than on the outskirts scattered around, giving them more power and ability to impose themselves. Remember, they were the only people allowed to have arms within Rome.
Senate- angered by Tib, and also feared to speak out. Often referred to as "men fit to be slaves". Tiberius wanted them to debate with him, but his own sullen characteristics and the idea of the maiestas trials caused them to fear stating any opinion contrary to his, worrying that it would be seen as treason. This was also a problem as there was no outlined definition of treason, so Tiberius could invoke it whenever he wanted. There were some times when the Senate went against him, and one time is noted that he went into the minority voting lobby and "not a soul followed him". Yet it seems that overall the Senate is subserviant, and Tiberius grew impatient with them. He also tried to maintain what Augustus set up, including with his Frontier policies.
The Maiestas trials, according to some Modern historians, were ramped up in the sources and were probably not as ruthless as they are described to be. The number of actual deaths due to Maiestas is also frequently debated, and is roughly agreed to be very few (plus some suicides due to the "honourable" way out).
His will was ignored, in part about the successor.
