Answer
$1~Btu = 0.252~kcal = 1056~J$
Work Step by Step
$1~Btu = (1~lb)(1^{\circ}F)$
$1~Btu = (1~lb)(\frac{454~g}{1~lb})(1^{\circ}F)(\frac{5^{\circ}C}{9^{\circ}F})$
$1~Btu = 252.2~g~^{\circ}C$
$1~Btu = 252~cal$
$1~Btu = 0.252~kcal$
We then convert from units of kcal to joules;
$1~Btu = (0.2522~kcal)(\frac{4186~J}{1~kcal})$
$1~Btu = 1056~J$