FPGA-based genetic algorithms (GAs) can effectively optimise complex applications, but require extensive hardware architecture customisation. To promote these accelerated GAs to potential users without hardware design experience, this study proposes a general-purpose automated framework for creating and executing a GA system on FPGAs. This framework contains scalable and customisable hardware architectures while providing a unified platform for different chromosomes. At compile-time, only a high-level input of the target application needs to be provided, without any hardware-specific code being necessary. At run-time, application inputs and GA parameters can be tuned, without time-consuming recompilation, for finding further good configurations of GA execution. The framework was tested on a high performance FPGA platform using nine problems and benchmarks, including the travelling salesman problem, a locating problem and the NP-hard set covering problem. Experiments show the system's flexibility and an average speedup of 29 times over a multi-core CPU.