Were there a massive increase in excess death in Australia in 2022? Yes, there was.
Was it 5,162%? Not even close.
Was it caused by COVID-19 vaccinations? No.
The Provisional Mortality Statistics for 2022 were released by the Australian Bureau of Statistics on Feb 24th, 2023. [I note that a new update is due this week; that link may have updated data by the time you read this.]
In 2022, there were 174,717 deaths that occurred by 30 November and were registered by 31 January 2023, which is 22,886 (15.1%) more than the historical average.
15.1% is both a massive increase and nowhere near the reported 5,162%. [Given a baseline average of 40 deaths per 100,000 Australians, a 5,162% increase would basically mean that 1 in 50 Australians died last year.]
Note: The provisional numbers cannot directly calculate excess deaths; further analysis is required, as the caveats of the document explain. These numbers are provisional and only rough.
Note: A comment from @user253751 proposes a different interpretation of the 5,162%. If true, the number is misleading - a 5,162% increase could indicate that the number of deaths has dropped dramatically below the long-term average. If this interpretation is the one intended, the wording of the linked source is misleading and their figures are largely gibberish.
Deaths with COVID-19 identified as the cause of deaths account for a large amount of the excess death. Deaths with COVID-19 as a contributing factor accounted for some more.

Note: This graph's Y-axis does not extend to zero, which may exaggerate the size of the difference to readers unfamiliar with that.
The document also identifies increases in deaths due to Alzheimer's disease, diabetes, and in increase in chronic lower respiratory diseases but a decrease in influenza and pneumonia.
These figures were widely reported and subjected to further analysis that explored the reasons further:
SBS: Australia recorded 20,000 more deaths than expected last year. This is why
The excess mortality is largely due to the coronavirus pandemic.
Actuaries Digital: COVID-19 Mortality Working Group: Almost 20,000 excess deaths for 2022 in Australia
- Our expectation is that excess mortality for the full year 2022 will be 12% i.e. there were nearly 20,000 more deaths in 2022 than would have been expected if the pandemic had not happened.
[...]
- Just over half of the expected excess mortality for 2022 is due to deaths from COVID-19 (+10,300 deaths), with another +2,900 where COVID-19 was a contributing factor, and the remaining excess of +6,600 with no mention of COVID-19 on the death certificate.
- While most of the excess deaths are in older age groups (i.e. 65+ years), excess mortality is a significant percentage in all age groups in 2022.
Most states and territories have had broadly similar levels of excess mortality in 2022, but the delayed re-opening of WA’s borders is reflected in a delayed uplift in mortality.
Sydney Morning Herald: Almost 20,000 more people died last year – the highest excess death rate since WWII
“Part of the explanation could be that people are going to hospital and not being able to access treatments because hospitals are full of COVID patients. At the peaks of COVID, ambulance and emergency department waiting times are high. We think that’s one factor in why there’s more non-COVID deaths coming through,” Cutter said.
“The other aspect of people not getting medical treatment could be they didn’t get the treatment they needed or the screening tests they should have in 2020 and 2021, so they’re having worse health outcomes that have taken a while to emerge.”
The linked article also claims:
The obvious culprit is, of course, the mass injections which in Australia were essentially forced upon people by locking unvaccinated people out of basically every public place with camps built throughout the country.
As an Australian who lived through the pandemic, I find this characterisation laughable, but I won't address it here. It needs a separate question.