In this paper we study the thermodynamic performance of a Brownian heat engine, which is driven by temperature difference. Brownian particles move in a periodic single-barrier sawtooth potential under the action of an external load force, and contact an alternating hot and cold reservoir. The kinetic energy change of the Brownian particles and the heat leak between hot and cold reservoir are considered simultaneously. The dynamics of the Brownian particle is governed by the Smoluchowski equation. The expressions for heat flow, power output and efficiency are derived analytically. The influences of the height of barrier, the external load force and the heat reservoir boundary on efficiency and power output are discussed by numerical calculation. It is shown that the Brownian heat engine is irreversible when the irreversible heat flow caused by the kinetic energy change and the heat leak exist. The curve of the power output versus the efficiency is a loop-shaped one. When the barrier boundary coincides with the heat reservoir boundary, the heat engine has a maximum power output. The efficiency of the heat engine is enhanced, but the power output is reduced when the heat reservoir boundary is changed.