I have the ICSP header (MOSI,MISO,SCK,Reset, GND, VCC) on the board. But you can only use this to upload a bootloader?
No, you can use it for any program. In fact, when you are using it to program the bootloader, the bootloader is just like any other program.
But I do use the exact same pins to program it via arduino uno (isp)?!
yes, the SPI pins, plus Vcc, Gnd, Reset.
What do I then need the RXD and TXD pins for (besides serial communication)?
You don't need them at all for ISP programming. They are only used by the bootloader to program if you have a bootloader loaded on the chip.
PS @pico: I do not own a USBasp or something like that but the arduino with the ISP sketch does the exact same thing, doesn't it?
Yes, but much less conveniently. You were asking what the "normal" way to do it was, the normal way is to use a real ISP programmer with a 6-pin header cable that fits onto your board's ICSP header. Quick and straightforward as it gets in that case.
The "Arduino with the ISP sketch" is a lot of futzing around by comparison. But yes, ultimately, you should be able to get it to do the same thing.
Be aware that whenever you program via a ISP programmer, if you have a bootoader program loaded, it will be overwritten, just like any other program (because to the ISP programmer, it is just any other program.) This only matters if you intend to do both types of programming on the same chip, but if you do, you need to burn the bootloader program on there again before you can program a sketch by RX and TX using the bootloader method.
Generally, though, you will be using one method or the other for a particular project.