The band kicked off the official Eurovision build-up on Friday, watched by fans on Liverpool's waterfront.
The celebrations will continue until the grand final on Saturday, 13 May.
Kalush's Tymofii Muzychuk said the band members were sad it was impossible to hold this year's Eurovision in Ukraine.
The winning country normally hosts the contest the following year, but organisers decided it was too dangerous to stage the annual extravaganza in Ukraine following Russia's invasion.
Instead, the UK - which came second with Sam Ryder in 2022 - is hosting this year's event on behalf of Ukraine.
"It's good that it's being staged in the UK," Muzychuk continued. "We see lots of Ukrainian colours, and the main thing is safety."
The musician told BBC News he wanted this year's contest to send a message to the world to not forget about Ukraine.
"The war in Ukraine is continuing," he said. "It's not finished yet and we want to remind people that they shouldn't lose track of it and it should be in the headlines."
Many people in the country will be watching the contest, he added. "They'll be supporting and rooting for Ukraine.
"Of course we are sad that it's not being staged in Ukraine but I hope that the UK entry wins, then we can swap and hold Eurovision in Ukraine [in 2024]."
Some fans waved Ukrainian flags as they watched the band, while others wore pink bucket hats - following the trend started by frontman Oleh Psiuk.
Kalush Orchestra will also perform at the grand final, opening the show with a performance titled Voices of a New Generation.
As well as the final and semi-finals, Liverpool is staging many other events around the city, including daily shows at the 15,000-capacity Eurovision Village.
On Friday, a special supergroup featuring musicians from Merseyside and Ukraine performed after Kalush Orchestra under the banner Welcome to Eurotopia.
The UK members included Orchestral Manoeuvres in the Dark singer Andy McCluskey, singers Jane Weaver and Natalie McCool and art-pop group Stealing Sheep, while Helleroid, Krapka;KOMA and Iryna Muha represented Ukraine.
Friday also saw a street parade as part of a a Eurovision-themed cultural festival. The Blue and Yellow Submarine Parade was inspired by the colours of the Ukrainian flag along with the song and film by Liverpool's most famous musical exports, The Beatles.
* Sunday, 15:00 BST - All 37 Eurovision entrants will be officially presented when they walk down a turquoise carpet
* Sunday, 20:00 - The National Lottery's Big Eurovision Welcome featuring a reunited Frankie Goes to Hollywood plus other local heroes the Lightning Seeds, Atomic Kitten and Ricky Tomlinson, and Eurovision legends Jamala and Conchita Wurst
* Monday, 18:30 - Highlights of both the turquoise carpet and welcome ceremony on BBC One
* Tuesday, 20:00 - Semi-final one
* Thursday, 20:00 - Semi-final two
* Saturday, 20:00 - Grand final. The semi-finals and the grand final are live on BBC One and Radio 2, with full coverage on the BBC News website
* Saturday - The Royalty Ball celebrates the north-west's vogue scene, plus a performance from the Scissor Sisters' Jake Shears
* Sunday - Family favourites Hey Duggee, Bluey and Andy and The Odd Socks, before emerging local performers
* Monday - This is Ukraine presents Antytila, Go_A, Alyona Alyona and Jerry Heil
* Tuesday - English National Opera gives Eurovision classics an operatic makeover with the help of singers Russell Watson, Ruslana - and Sonia!
* Wednesday - The Lightning Seeds, Miles Kane and Girls Don't Sync lead a line-up of Liverpool talent
* Thursday - Ukraine's 2016 Eurovision winner Jamala premieres her new album, inspired by Crimean folk songs
* Friday - EuroEve with Fleur East, Sophie Ellis Bextor and Charlotte Church's Late Night Pop Dungeon
* Saturday - Grand final day with Katrina from Katrina and the Waves, Jedward's Giant Eurovision karaoke and Claire from Steps
Entry to the fan village is free except for grand final day, which is ticketed and already sold out.