According to Wikipedia, Didius Julianus was proclaimed Roman Emperor in AD 193 by the Praetorian Guard after offering them 25,000 sesterces per soldier (outbidding Sulpicianus's offer of 20,000 sesterces each). But "25,000 sesterces" is just a number. How much money would that have been, compared to the yearly pay of a typical soldier (or a typical Praetorian)?
Didius Julianus (ruled 28 March to 1 June 193 AD) promised a total of around 200 million sesterces to the Praetorians but apparently did not have sufficient funds to meet this commitment. For context, a legionary foot soldier was paid (annually, before significant deductions) around 2,400 sesterces following the pay rise in 197 AD, and a praetorian an estimated 6,000 to 8,000 sesterces.
In answer to the main question, How much did Didius Julianus pay to become emperor?
There were an estimated 8,000 praetorians at this time so:
8,000 x 25,000 sesterces = 200,000,000 sesterces in total (or 50,000,000 denarii), though officers may have been offered more.
It was quickly discovered, however, that he had lied to the praetorians and deceived them, as he was unable to fulfill his promises.
The truth is that he did not have as much money in his personal possession as he had pretended to have, and no money was available in the public treasures;...
The lack of public funds (which Julianus might otherwise have used to make up for the lack of his own) seems to be supported by Cassius Dio who, when commenting on Julianus' immediate predecessor, Pertinax (emperor Jan-March 193 AD), noted that:
At this time, then, there was such a dearth of funds in the imperial treasury that only a million sesterces could be found.
The answer to How much money would that have been, compared to the yearly pay of a typical soldier (or a typical Praetorian)? is rather more difficult as documentary evidence is limited for this period.
The basic annual pay of a legionary foot soldier in AD 197, during the time of Didius Julianus' (AD 193) successor Septimus Severus (AD 193 - 211), is estimated to have been 2,400 sesterces. As there was a pay rise in AD 197, the figure for 193 AD would have been less than this, and there were also substantial deductions made for food, clothing, boots etc., but soldiering was nonetheless a well-paid occupation.
Michael Speidel, 'Roman Army Pay Scales'. In The Journal of Roman Studies (Nov. 1992)