Hi, it is possible to slightly upgrade the CPU, but improvement in gaming and streaming performance would mostly come from upgrading the video, which will likely also require an upgraded power supply.
First, the CPU options: A10-8750 should be compatible, but is only an extra 100 MHz. Other options are Athlon X4 860K. 870K, and 880K. These perform slightly better than the current processor, and use some more power. They also have no integrated video, and thus require a discreet video card. Any of the above 4 processors is a slight upgrade, but likely barely noticeable.
Next, graphics: the R7 in the A10-7800 offers 512 shader cores at 720 MHz, and uses system memory. Much more performance is obtained with a discrete graphics card. For example, RX580 offers 2304 stream processors at 1120 MHz, and uses 4-8 GB of dedicated video memory. Performance should be sufficient to stream.
Finally, the Lenovo 90BG003JUS tower has a 180W power supply, which is insufficient for a discrete graphics card, so upgrade to a quality 500W+ psu.