test.js

function ab2str(buf) {
  return String.fromCharCode.apply(null, new Uint8Array(buf));
}

function str2ab(str) {
  var buf = new ArrayBuffer(str.length*2); // 2 bytes for each char
  var bufView = new Uint8Array(buf);
  for (var i=0, strLen=str.length; i<strLen; i++) {
    bufView[i] = str.charCodeAt(i);
  }
  return buf;
}


String.prototype.getBytes = function () {
  var bytes = [];
  for (var i = 0; i < this.length; ++i) {
    bytes.push(this.charCodeAt(i));
  }
  return bytes;
};

var client = forge.tls.createConnection({
  server: false,
  verify: function(connection, verified, depth, certs) {
    // skip verification for testing
    console.log('[tls] server certificate verified');
    return true;
  },
  connected: function(connection) {
    console.log('[tls] connected');
    // prepare some data to send (note that the string is interpreted as
    // 'binary' encoded, which works for HTTP which only uses ASCII, use
    // forge.util.encodeUtf8(str) otherwise
    // client.prepare('GET / HTTP/1.0\r\n\r\n');
  },
  tlsDataReady: function(connection) {
    // encrypted data is ready to be sent to the server
    var data = connection.tlsData.getBytes();
    var datab = str2ab(data);
    console.log("tlsDataReady", data, datab);
    socket.send(datab, function(writeInfo) {
    });
  },
  dataReady: function(connection) {
    // clear data from the server is ready
    var data = connection.data.getBytes();
    console.log('[tls] data received from the server: ', data, ab2str(data));
  },
  closed: function() {
    console.log('[tls] disconnected');
  },
  error: function(connection, error) {
    console.log('[tls] error', error);
  }
});
var HOST = "192.168.1.111";
var PORT = 8009;
// var HOST = "google.com";
// var PORT = 443;
var socket;
chrome.socketsExtended.tcp.open({}, function (_socket) {
  socket = _socket;
  socket.on("receive", function (msg) {
    console.log("receive", msg);
    client.process(ab2str(msg.data));
  });
  socket.connect(HOST, PORT, function(result) {
    if (result < 0) {
      console.log('[socket] disconnected');
    } else {
      console.log('[socket] connected');
      client.handshake();
    }
  });
});

var buf = new ArrayBuffer(str.length*2); // 2 bytes for each char

You don't need 2 bytes for each char. The string's internal implementation will use two bytes to store each character in a binary string, but each character code is < 256 and when converting to a Uint8Array the number of bytes will match the number of characters.