The original spelling in Polish is Kleczewski, pronounced roughly "kleh-CHEFF-skee." So Kleczewski is a very reasonable phonetic rendering of the name by English values. It was not unusual for Polish immigrants to go along with slight changes to their names' spelling, to make them easier for non-Poles to pronounce. Americans would look at Kleczewski and probably pronounce it something like "kleh-ZOO-skee." By spelling it Klechefski, the family managed to retain the original pronunciation, more or less.
The late Polish surname expert Prof. Kazimierz Rymut mentions this name in his book Nazwiska Polaków [The Surnames of Poles]. He says it usually originated as a reference to the name of the town Kleczew, which you can see here: