One of the reasons for this critical designation as a poet of failed relationships is poems about failed relationships, like "Daddy." This symbolic poem ("Daddy, I have had to kill you. / You died before I had time---") epitomizes the failed relationships in Plath's life and certainly contributes to her representation as a poet of failed relationships.
I must agree with the other postings. Plath was known for her poetry in regards to how it mirrored her life and her mental instability. She was, historically, known for her failed relationships, depression, and, ultimate, suicide. Plath is simply, wrongly or rightly, stereotyped as the poet whose work showed her very intimately and honestly.
#2 makes a number of excellent points. I would merely add that I think one of the reasons why Plath's poetry is so raw and great is precisely because of the way that it mirrors so many of the experiences she had in life. The emotional intensity of poems such as "Daddy" and "Lady Lazarus" is a direct result of the pain, hurt and broken relationships that she experienced in life. If we characterise her as a poet of failed relationships, this is no way denigrates her work or her achievements.
I think that Plath's poetry does seem to be affected by her relationships. What poetry isn't? If we are labeling her, it is a label that many people would attach. I think most people consider her name synonymous with suicide and depression. There is more to her poetry than that.
Plath had failed relationships. This is perhaps the most honest reason why she is considered a poet of failed relationships, it happens to be the truth. However, people would not have been so aware of her relationships had they not known about her suicide, and then further prodded into her life. Here are a couple other reasons to consider.
1. CONTENT: The content of many of Plath's poems plays on rich and specific raw emotions. She did this through crafting specific images and symbols with carefully chosen words and figures of speech. Generally choosing everyday experiences and items to write about, she did indeed write about some of the pain in her life including the "The Jailer" about her husband Ted Hughes who had cheated on her, and "Daddy" about her father who abused her. These topic choices come from truthful circumstances, but readers generally really sympathize with Plath because of her suicide and what led her to it. So, we readers in turn often see even more than what she intended for us to see.
2. INABILITY TO SEPARATE SELF FROM WRITING: Plath could not take herself out of her writing. She could not write outside of her own perspectives or circumstances. She knew this about herself and was highly criticized for it. This means that she failed to be objective in several instances.
A few good poems to use to capture her struggle with relationships might be these: