How can players with 10 victories be level 25 in League of Legends?

Lets look at what is required to get to level 25+

Chart Source

The received Experience Points are calculated at the end of each game; wins are worth more than losses, and the longer the game lasts, the more XP you gain.

Assuming an average XP gain to be about 110 XP per normal game, one can reach summoner level 25 within 246 normal games. Using XP boosts will double the amount of XP you recieve. So, lets assume someone got the maximum XP which for winning a normal game will be around 200 + the use of XP boosts = 400 xp per win. One can get to 25 in 68 games. Source

However, it is important to note that depending on where you are looking will depend on what information is shown to you. For example, if you are seeing the 10 wins from a summoner at the end of a normal game then it will only show the stats from winning normals. This means it won't show any stats for winning any other mode (Dominion, ARAM, CO-OP vs AI, custom games).

It is quite possible that someone could reach 25 through playing only Dominion, ARAM etc games and has only played around 20 games of normal. For you to get a real idea of how many games someone has played I would recommend using lolking.net. It should show all the games someone has played in which mode, their wins, losses and many other statistics.

There are basically two ways to achieve this:

The first would be by just playing other Queues than normal 5v5. Other Game queues aren't listed as "wins" and thus the low amount of won games in normals.

The second method is rather weird and requires him to really be losing all the time. This would require a incredible high amount of games for him to lose which is highly unlikely to happen.

Either way there is no "hacking" or "exploiting" involved.

You pretty much answered your own question.

They could be playing with bots, losing a lot, or perhaps even using experience boosts.