What will the last day of humanity look like? It's certain that human life on Earth will not go forever. Therefore there will be one last generation of humans on Earth. 


How will life look like for last few generations approaching the end?
How will life end for the very last people, will they all die at once as if struck?


Assume the technology won't be much more advanced than today, no colonized planets, and we'd stay biologically very similar to now. Also assume that no meteor will struck us and end everything but that extinction will happen due to aging of the Earth and Sun.
The last humans in this scenario likely die of starvation and/or drought during a natural short-term climate fluctuation. 

The basic issue is that as the sun heats up there will be a gradual reduction of carbon dioxide in the atmosphere (due to plant life but also silicate weathering, which speeds up in warm environments). There may also be a reduction in volcanic outgassing, although I think the timeframe is too slow. This is bad for plants, since they will get less CO$_2$ to photosynthesise from, and it will reduce the climate stabilising feedback so temperatures will start going up as the sun increases in luminosity. Eventually we get to a runaway greenhouse stage where water vapour alone triggers overheating, but this is possibly long after the biosphere is mostly dead.

The most complete model of this I have seen is this paper by Franck, Bounama, &amp; von Bloh. They estimate complex multicellular life to go extinct in 0.8–1.2 Gyr.

Note that this will happen long before the sun becomes even a subgiant. 

In any case, if this model is correct we should expect that slowly but surely the future humans would be forced to migrate to cool regions and would be especially limited by the dwindling biomass available. But beside this they will be subject to normal random climate fluctuations that sometimes create decade-long droughts or other misfortunes, and it is likely one of those that actually gets the last individuals.

Of course, the scenario assumed makes some very strong assumptions. In order to remain recognisably humans over millions of years these humans would have to engage in deliberate genetic control, otherwise genetic drift will make them mutate into new species. It assumes that nobody does anything about the long-term trend. And so on.