Taking the tube from Heathrow (Picadilly line) to Kings Cross station: Can anyone tell me how difficult (i.e. stairs, distance) it will be with luggage? I am trying to make a train connection to Leeds on a Thursday afternoon around 3PM.

Thousands of people do similar every day. A lot depends on how heavy and how many bags you have with more bags being very bad

Basically one 24" roller and a carryone which fits on top. Books take up most of the weight. Are there lots of stairs there? Escalator? Elevator? Is it similar to Russell Square where I thought I climbed up at least 1,000 steps. I was hoping that since this was a train station there would be some consideration for travelers? Obviously I have to do it but wanted to know if I needed cross-country training before taking the trip.

Kings Cross railway station, especially if you're taking a long distance train, is extraordinarily small: it's practically impossible to walk more than 50-100 yds from the tube exit even to the farthest end of the longest train.

To get to it from the Piccadilly line is almost entirely escalators, though when the tube station's very busy you might sometimes be pulled into a crowd management system that routes you into an extra 100 yards' walking - all on the flat. Because the tube station is vast, it's important to keep following the signs for Kings Cross railway station. There's a 12-step climb just before the railway station itself, with a nearby - but poorly signed - lift if you don't want to use the steps. For almost everyone, lugging luggage up the steps is a great deal less hassle than messing with the lift, and I'm honestly not sure how easy it is to spot the lift, or get to it, if you're in a flow of people heading up the steps at the busiest times.

Incidentally, I've never used Russell Square station when it's been necessary to climb up the steps. Just once in a longish life I've been in the tube system when all the Russell Square lifts were out of action. Warnings to avoid the station were being broadcast in every station and on almost every train.

Thank you Flanneruk, that was exactly the type of information I was asking for. I may have been slightly exaggerating the stairs at Russell Square. As for the warnings being broadcast at the station, I am an American and sometimes it is difficult to understand your English. It took a little while to comprehend whatever you mean by "mind the gap"; a large host of horrors vaguely having to do with dental problems was my immediate suspicion.