It is correct and can be used in written English.
You can use it to refer to a series of accomplishments. For example: "John has had a string of achievement in his professional career, ranging from earning multiple PhDs to running a successful business."
Wanda Jackson's accomplishments could line a highway from here to Nashville -- a string of achievements that includes being the first woman to record a rock song.
This puts pressure on whichever political leader happens to be in the chair not merely to preside efficiently, but also to notch up a string of "achievements".
It is certainly true that Mr. Clinton in his instincts and messaging was more centrist than Mr. Obama, and the 42nd president emerged from the White House with a string of achievements that both parties laid claim to.
It may have been those qualities — and a string of achievements that began in her native North Dakota, where she was state student council president, an all-state basketball player and orchestra concertmistress — that drew the attention of David H. Petraeus, the nation's top spy and a four-star general, as the two spent hours together for a biography of Mr. Petraeus that Ms. Broadwell co-wrote.
Getting into a selective graduate program typically involves a string of achievements – I'm sure there's a come-from-behind story somewhere in there.
They have been successful rule-followers since childhood; cooperative and compliant, they have responded to the high expectations of parents and teachers with a string of achievements.
And he has a string of legislative achievements to his name.
Just a few days ago, the 107th Congress was racking up a string of unexpected achievements on corporate responsibility and trade despite divided party control.
It has produced a string of scientific achievements: confirming that most galaxies have a black hole in the middle; providing a front-row seat for the collision, in 1994, of a comet with the planet Jupiter; and helping to uncover the strange fact that the expansion of the universe seems to be accelerating.
