Simply pick a spot and use any rod.
These actions will cause the chain to break*:
Reeling it in too late/early
No fish biting (use a Pokemon with Suction Cups / Sticky Hold ability in the first party slot to reduce the chance of this)
Loading/reloading a map
Entering a building
Ending the game / switching the 3DS off
These actions will not cause the chain to break*:
Running from battle
Catching/Fainting the Pokemon
Fishing up a different Pokemon
Using Potions / Repels / PP restore from menu
Using a different fishing rod
Equipping the Bicycle / Dowsing Machine / Skates
Moving to another spot on land (within the same map)
Encountering other Pokemon in grass.
Crossing a map boundary as long as it does not involve doors and/or loading a map. (e.g. it is okay to move along the coast of Cyllage City, Lower Route 8, Ambrette Town)
Using Surf and Waterfall
Moving from land to water using Surf
Moving on water
Encountering Water Pokemon by moving on water
Using Rock Smash
Encountering Pokemon from the Rock Smash
Saving the game (but this doesn't matter because the chain breaks if you switch off the game, so you can't load a save file that maintains the chain).
Closing the 3DS lid (sleep mode)
Pressing the home button (suspended game)
*A Summary on how this information was verified:
Started with a fishing record of 5 consecutive.
Fish 6 consecutive times
Perform the alleged action that will break the chain
Fish another 2 consecutive times
Check against the fisherman in the fishing shack on Route 16
If he reports 6 as the longest chain, then the action that was investigated breaks the consecutive chain. If he reports 8 as the longest chain, then that action does not break the chain.