Paweł Kochański was born in Odessa, then part of the Russian Empire, to Polish-Jewish parents and studied violin first with his father and then at age 7 with Emil Młynarski, whose teacher had been Leopold Auer. In 1898 Młynarski went to Warsaw, and upon founding the Warsaw Philharmonic Orchestra in 1901, summoned Kochanski, then aged 14, to be its concertmaster. He also took charge of his upbringing and education, treating him like a son and stating that he believed he would become a world-class soloist. In 1903, with sponsorship from leading Warsaw families arranged by Młynarski, Kochanski went to Brussels to study with César Thomson at the Brussels Conservatoire. There, after four months, he received the Premier prix avec la plus grande distinction (First prize, with the greatest distinction).
It was at this point, as he was beginning his itinerant virtuoso career, that he met Arthur Rubinstein, through the invitation of Juliusz Wertheim. They immediately realised their shared musical sympathies, but the friendship, rich with youthful energy, really took off in 1907 with their concerts for the Warsaw Philharmonic, including duo performances of Beethoven's Kreutzer Sonata and Tchaikovsky's Piano Trio with the cellist J. Sabelik. In 1908, with Jozef Jaroszyński (a patron of Kochanski's), they made a triumphant tour of European capitals, including Berlin, Paris, London and Karlsbad, and in 1908–9 Kochanski and Rubinstein performed the Franck Violin Sonata in A, the Kreutzer again, and a Brahms trio (with Eli Kochański, a cellist and Paul's gifted brother) for the Warsaw Philharmonic.
From 1909 to 1911 Kochanski taught at the Warsaw Conservatory as professor of violin. In 1909 he and Rubinstein gave the first performance of Karol Szymanowski's Violin Sonata in D minor. Their participation, with their friend Szymanowski, in the movement known as Young Poland, helped to promote more progressive musical attitudes in Warsaw. In 1911, Kochanski married Zosia Kohn (who had previously held a hopeless passion for Juliusz Wertheim). His father-in-law, a lawyer, bought him a Stradivarius violin for his wedding present. Szymanowski dedicated his Violin Concerto No. 1 in 1916 to Kochanski, who contributed the cadenza.
In 1913–1914 in London, Rubinstein introduced Kochanski to the music room of Paul and Muriel Draper, to which they also introduced Szymanowski, and where Paul met Igor Stravinsky. In this circle they were often with Pablo Casals, Jacques Thibaud, Lionel Tertis, Pierre Monteux and others. Stravinsky dedicated a transcription for violin and piano of three pieces from The Firebird to Kochanski, who participated in two of Rubinstein's recitals at Bechstein Hall in 1914, one of which was devoted entirely to contemporary music.
In 1916 he succeeded Leopold Auer, teaching at the St. Petersburg Conservatory until 1918; during that time he became friends with Sergei Prokofiev and gave the composer some assistance on matters of technique for the solo part of his Violin Concerto No. 1 in D major. He moved on to teach at the Kiev Conservatory from 1919 to 1920. In January 1920, he premiered Szymanowski's Nocturne and Tarantella in Warsaw.