The champion ski racer said that Bleakley had made the best of a "tough situation" that saw her replace fan favourite Holly Willoughby.
"Christine came up to me after I'd had a tough week and put her arm around me and talked to me for ages, which was lovely, and she's always in the bar after the show," Alcott told The Sun.
"Everyone loved Holly Willoughby, who Christine took over from as host, so I think that's a tough situation to come into, but she's been brilliant."
Alcott, who described herself as "an athlete, not a celebrity", went on to praise her famous co-stars for being approachable and grounded.
"What this show has taught me is that celebrities are just like normal people even though we sometimes put them on a pedestal," she said. "I've made really good friends with Jorgie Porter.
"I never thought me and actresses would be great friends but we've all become like a little family... everyone has been so kind."
Bleakley recently blasted "ridiculous" criticism of her Dancing on Ice hosting abilities, saying: "If two or three people are wanting to say something that is not nice, out of ten million watching, then that's fine. I was overwhelmed by the lovely comments I got and that's all I focus on."
> Dancing on Ice Katarina Witt sorry for Chemmy Alcott 'big woman' jibe
Dancing on Ice continues tomorrow (February 5) at 6.25pm on ITV1.
Watch Chemmy Alcott on Dancing on Ice below: