First of all, it must be clear that there isn't just one chain: every node in the network has its own local chain stored locally. Through message exchanges, all the nodes chains would eventually end up synchronized.
Now imagine the situation in which there are no miners and Alice wants to send 1 BTC to Bob. She sends the transaction to a node in the network and the node updates its local chain with the new transaction.
In the meantime, Alice sends the very same Bitcoin to Charlie but this time she notifies a different node of the network with a new transaction, and that node will properly update its local chain with the new transaction. Now both nodes have two different transactions in their local chain. They start sending messages to their neighbour nodes telling that new transaction have been arrived. The nodes will receive updates from both the nodes but will not be able to discriminate which is the legit spending.
Mining is there because in this way all the nodes can agree on the same set of transactions: finding a solution to the hashing problem will guarantee (at least, most of the time) that all the nodes chains agree on the same set of transactions. This happens because only the one who finds the Proof of Work can add and propagate a block in the network. Finding the Proof of Work is a hard task, and due to its randomness nature it is probable that only one miner will find a Proof of Work.
By the way, this question has been asked several times, i suggest you to take a read from the official Bitcoin Wiki that will enlight you on the features of a blockchain.