The Rema (565:5) citing the Ohr Zoruah writes, if one was unable to recite selichos before davening with a minyan, he shouldn’t recite them later on b’yechidus. However, the Mishnah Berurah (565:13) quoting from Achronim asks on the above, and doesn’t understand why one shouldn’t recite selichos in private as they are just tachanunim [supplications]? The Mishnah Berurah concludes that one can say selichos in private, however, he should miss out the Yud Gimmel Middos.
In explanation of the Rema, there is a machlokes Magen Avraham and the Taz. The Magen Avraham (6) explains, that selichos used to be recited during Shemonah Esrei, therefore, the Rema held that when davening in private one shouldn’t say them, as it creates a hefsek [interruption]. However, the Taz (5) explains, that the reason one shouldn’t say selichos in private, is because even if one misses out the Yud Gimmel Middos, there are still many other parts which also need to be skipped out, such as, "the merits of the Yud Gimmel Middos" and many other parts of selichos which mention the merits of the Yud Gimmel Middos. However, certainly parts which make no mention of the Yud Gimmel Middos, should be said, even when one is davening by himself.
The Mateh Ephraim (581:21) and Kitzur Shulchan Aruch (128:9) also write that one may recite selichos in private, however, he should skip any reference to the Yud Gimmel Middos. However, from the Mishnah Berurah it seems that one may recite everything except the Yud Gimmel Middos itself, and it’s ok to recite the pieces which reference the Yud Gimmel Middos, even when davening alone (see however, Mishnah Berurah 581:4).
