Heathrow to Kings Cross

Piccadilly line all the way (dark blue on the tube map), no changes. Takes about 50 minutes.

It may be slightly quicker via Heathrow Express to Paddington, and then the tube, but it will cost more, and it is up and down stairs to change to the tube at Paddington.

That is, as long as you only have a small amount of luggage. If you can't carry your luggage on and off a tube train, then it is Heathrow Express plus taxi.

Although I don't disagree with the choice of Piccadilly all the way, the route from the Heathrow Express to the Hammersmith & City Line is now step free as a lift and ramp is now available.

The Piccadilly is also step free at both ends of the Heathrow - Kings Cross journey, if you look for the lifts at KX.