You’ll need to escape the backslashes \ when putting them inside a js string literal.
<script>
data = '{"type":"MESSAGE","data":{"topic":"channel-bitsevents.XXXXX","message":"{\\"user_name\\":\\"TestUsername12345\\",\\"channel_name\\":\\"TestChannel12345\\",\\"user_id\\":\\"TestUserID12345\\",\\"channel_id\\":\\"TestChannelID12345\\",\\"time\\":\\"2017-06-3T12:00:00.000Z\\",\\"chat_message\\":\\"cheer1 Test message.\\",\\"bits_used\\":1,\\"total_bits_used\\":1,\\"context\\":\\"cheer\\"}"}} ';
console.log(JSON.parse(data));
</script>