How to remove inactive members from clan?

Inactive people can easily be spotted by either visiting their village or checking whether they are in a league.

Most no league people are inactive unless a new season just started. In that case, check their villages. Inactive bases will have lots of tombstones and full mines and collectors. Also, if no one sees them in chat, as to talk or to request for troops, they are inactive. Lots of bases though will have tombstones due to recent raids and full collectors and mines due to the fact the user hasn't come one yet for about some time.

Of course, removing them would be kicking them out: click on their name after clicking the Clan button (seen as an option when the Clan Castle is chosen) and click "Kick Out". You can leave them a little message or just keep them as clan advertising then kick them out.

If you are leader you can go to your profile(the little "i" next to your experience bar), then click the "My Clan" tab, then tap on the name of the person you want to kick out and a menu will pop up, then simply tap the one that says "Kick Out".

Too verify inactive members there is no easy way to do it, but if they have 0 attack wins, 0 donations, and 0 received donations, they are probably inactive provided it isn't the beginning of a season. This especially true if you check near the end of a season. Another thing to check is collectors and builders. If collectors are usually close to full and builders aren't working, it can also help tell if they are inactive.

As for running a clan it depends on how you would like to do it. Personally if they don't listen, I just kick them. Having a "good" member that doesn't listen isn't helpful. Constructive criticism, or advice is appreciated, but in the end it is the leaders clan, so he has final call.

Note: Co-Leaders can kick elders/members.

I would do this in a different way than the other posters. The best way to check someones activity is to ask the members. Since you have so many, that may be difficult, but spend a day or two looking at clan chat often and ask everyone to mention if they are active of not. The people who don't respond in 2-3 days should be marked as inactive, but don't kick them yet.

Ask if anyone in the clan knows them, since they might be on vacation, or friends of someone in the clan who can verify their current situation.

Finally, as mentioned by other posters, assuming it is not a brand new season, check their league, number of attacks and donations, with donations probably being the most important. Obviously, if it's 3 weeks into the season and they don't have a league and haven't responded to any of your queries in clan chat, they are probably inactive. The last thing to do before kicking them (if you really want to verify) is to check their base for destruction/tombstones/full collectors.

Unfortunately, you may end up kicking a few people that don't deserve it, but that's just how it goes as a clan leader. They can always rejoin/reapply later if they are really interested and put in their message that they were kicked incorrectly.