We report an efficient Cu K(alpha) x-ray source produced by focusing submillijoule, 120 fs Ti:sapphire laser pulses on a solid copper target to a spot diameter of few microns. The experimental results show strong emission of K(alpha) x-rays from solid targets from microplasmas created by p-polarized 0.2-0.3 mJ laser pulses at 1 kHz repetition rate. We have demonstrated K(alpha) x-ray point source emission rates of 6.7 x 10(9) photonss into 2 pi sr at 1 kHz repetition rate. The source has an x-ray conversion efficiency into Cu K(alpha) line emission of 3.2 x 10(-5). The source has a measured size of approximately 8 microm. Such a high repetition rate K(alpha) x-ray source can be very useful for time resolved x-ray diffraction and radiographic applications.