As a fellow laptop user who had some issues regarding his fps over the past few months, here are some suggestions which may eliminate the problem.
Double check your graphical settings in League. Set everything to low, and maybe take down the resolution a bit?
Power plans on Windows can control the maximum clock speed that your processor can reach. Ensure that your laptop is plugged in, and head to Control Panel -> Hardware & Sound -> Power Options and click the High Performance power plan. This will allow your processor to be clocked to its highest speed at the cost of extra power consumption (which is why you'll need to have it plugged in).
Laptops, especially lower-end ones, suffer from limited ventilation and the parts can easily heat up much more than their desktop variants. This heat can cause your pc to automatically down-throttle your cpu/gpu until the temperature has been lowered to a certain degree. I recommend using a CPU temp monitor, such as Core Temp, and observing it while you play games.
Background Programs & Services
Windows 10 has an odd habit of flooding your memory and disk usage with a ton of annoying background programs and services. These can affect the performance of just about any other program you're running, especially games.
First, I recommend using ShutUp10 to disable many background tasks that Windows might perform.
Open Run (Win Key + R), type services.msc and hit enter or ok. Head down to Superfetch. Right click -> Properties, and set the Startup Type to Disabled.
While running LoL, open task manager, ensure more details are shown, and then click the CPU, Memory and Disk headings, in that order. Take note of any programs that are using a high amount of CPU/Memory/Disk. If possible, close them, though be careful not to close any system crucial programs.
If at all possible, get some hardware upgrades, such as more memory, or an ssd (or both if possible). Currently you have 4GBs of memory, while 8GB is highly preferred nowadays. Also, a SSD will virtually eliminate processes that tend to hog the disk usage and slow your computer down. It will also drastically minimize any load times you have across your system.
Hope that one of these will help. Lemme know how it goes.