Is it possible to program AVR microcontrollers using USB-UART converter instead od USBasp?
AVR microcontrollers are programmed using either the ISP (In-System Programming) interface or JTAG (Joint Test Action Group, thats an IEEE standard for in-system programming and debugging of microcontrollers, FPGAs, ...). AVR ISP uses a three-wire SPI (Serial Peripheral Interface) to communicate with the microcontroller and further has to be able to drive the reset pin. SPI and UART are very different (SPI is synchronous while UART is asynchronous), so you can not use a simple USB to UART converter to program your microcontroller

An alternative to this is the use of a bootloader. The bootloader is a part of software on your microcontroller which allows programming via e.g. USB or UART. The problem is that you have to program the bootloader using ISP or JTAG before you can use it. Many evaluation boards come with preinstalled bootloaders to allow programming via USB. 

If you can find somebody with an ISP programmer, they could program a bootloader and you can then use the USB to serial converter. Otherwise you will have to buy/build an ISP programmer or switch to a controller or eval board with a preinstalled bootloader.