I want a darker handle when this is done but I still want to see the grain of the wood.
a light sanding with a very fine grit paper or steel wool should take away the grime.
i have not used teak oil to comment on how it would work out. but i wanted to point out that it's possible to get a lighter finish than the handle shown above. depending on what you consider to be "darker". it is possible to go so dark that the grain isn't very visible.
using a clear poly from minwax will also show the grain, but the handle will have a more natural color. both would look good. a darker finished handle will add contrast and make the kettle's lighter color pop, light will make the handle blend in more overall. it's really a matter of preference.
putting just clear over that handle once sanded will have this effect.
but if you want it darker then go the oil route. i have used poly finishes for some time, and they do hold up to weather very well so i don't think you'd go wrong either way.
good luck and keep us posted!