While the chess scene is generally focused on developing cinematography skills, the chase scene is mostly about editing. However, bear in mind, as ever, that film is a synthesis of story, sound and image, so in reality everything is assessed all the time!
Start where we always start; MAKE A PRODUCTION SCHEDULE.
Then, learn about the thing you are trying to make by watching as many examples of it as you can find. At this stage, watch professional rather than student examples. Here's a start; some very well known examples. Watch ACTIVELY - make notes, look for tricks and techniques and the conventions which pop up repeatedly, start forming your own ideas.
There are certain points of technique and narrative which you should notice. There are more examples here. One student has done a good job of identifying some commonly used tricks associated with chase scenes.
So, obviously you need to plan for the edit. Think of good locations, go there to check them out, storyboard the key shots and plan to shoot from multiple angles. If you want to introduce some motion into your cinematography, decide how you want to do that (sliders? Hand holding? Stabilisers? Shoulder mounts? Jibs?) and block out those shots.
Note, however, that all of this is still not enough. The best chase scenes are narratives in themselves; they have peaks and troughs of excitement, things happen to change the situation, the audience is misled and wrongfooted. Watching someone chase someone else is actually quite boring after a few seconds; you need to get a screenplay writer to impose some narrative on it. Think in the usual narrative terms - we need conflict and enigma, location, characterisation, equilibriums and disequilibriums.
See here for a good student example.
Many students sacrifice quality for quantity when it comes to chase scenes; they film people running fairly randomly (and unwillingly, a lot of the time) and assume they'll 'fix it in post.' Better students, of course, take the opportunity to plan really beautiful shots and give their editors something wonderful to work with. As ever, planning is everything.
Presumably, most of you want to work within the Institutional Mode of Representation - the invisible editing style so familiar from Hollywood-style chase scenes. In order to give your editor the chance to create those all-important match cuts, you need to be filming for continuity. Expect to film your shots several times! See the opening shots (here the girl bumps into the guy at the start) in this example - how many different angles? And then watch the rest of it to see how music, narrative, image, editing and performance all work together. And then watch the one beside it to see real attention being paid to shot, edit, sound, narrative. Remember your music also, should you want it; it should also be planned and produced at this time.
Hopefully, you have worked efficiently and well and given your editor enough time to edit and to demand pick-up shots if (when) they are needed. Get the sound AND the footage to the editor as soon as possible, because they need to work together.
Most people, as I said, will want to create very slick, invisibly-edited pieces. But we always have editors working in the Russian formalist tradition, either totally or partially. If you have one of these rare creatures, let them do their thing!
Assessment criteria - the usual film portfolio criteria- can be found here.