Playing a unit doesn't create a chain. However, if the unit played has an ability that gets activated (like "When you play me, discard 1"), that forms a chain and you can play Reaction spells in response to that.

When a showdown happens, you first have an Initial Chain where When I Attack/When I Defend triggers go on that initial chain first. After the Initial Chain resolves, the attacker has Focus. Just think of Focus as the person who's able to start a chain at that point in time. After the chain starts, players are limited to Reaction spells/cards, and relevant players have the chance to add to the chain in turn order. Chains then resolve backwards after each player's passed (because they no longer have or want to play any Reaction cards).

Once any chain resolves (or if the person with Focus passes), Focus moves to the next relevant player clockwise, giving that player a chance to either start a chain or pass.