I can read neither Russian nor any Turkic language (if you can read any of them, I'd recommend to check the bibliography in the end section of Kawaguchi & Nagamine 2016 below by yourself), but at least in Anglophone and other European historiography we have difficulty in finding almost any modern biography of Mongol rulers, regardless of academic of popular history one, except for Chinggis himself and Kublai, I suppose.

This monograph on Qaidu (d. 1301) is an rare exception: Biran, Michal. Qaidu and the Rise of the Independent Mongol State in Central Asia. London: Routledge, 1997 (pbk 2016).

As for the period around 1260, such as Berke–Hulagu war, the following two recent overview book will shed some light:

• Favereau, Marie. The Horde: How the Mongols changed the World. Cambridge, MA: Harvard UP, 2021: focuses on rather Möngke-Temür than Berke himself, but it offers a readable overview of the Golden Horde from the 13th to the end of the 14th century.
• Jackson, Peter. The Mongols and the Islamic World from Conquest to Conversion. New Haven: Yale Up, 2017.

As for the Ilkhanate and neighboring powers around 1260, you can also check Reuven Amitai (-Preiss)'s work. The classic authority of the Mongol Empire, Thomas Allsen, also has a strong point in his research in the middle of the 13th century (he authors the monograph on Mongke: Mongol Imperialism: The Policies of the Grand Qan Möngke in China, Russia, and the Islamic Lands, 1251–1259, Berkeley: U of California Pr., 1987).

On the other hand, these books rarely describe the early life of Berke up to his ascension to the leader of the Horde.

Works mentioned:

• Kawaguchi, Tatsushi & Nagamine Hiroyuki. 'Rethinking the Political System of the Jöchid'. Acta Orientalia Academiae Scientiarum Hung. 69-2 (2016): 165-81. DOI: 10.1556/062.2016.69.2.4 (in English)

(Edited): fixes typos.

Hi there anyone interested in recommending things to OP! While you might have a title to share, this is still a thread on /r/AskHistorians, and we still want the replies here to be to an /r/AskHistorians standard - presumably, OP would have asked at /r/history or /r/askreddit if they wanted a non-specialist opinion. So give us some indication why the thing you're recommending is valuable, trustworthy, or applicable! Posts that provide no context for why you're recommending a particular podcast/book/novel/documentary/etc, and which aren't backed up by a historian-level knowledge on the accuracy and stance of the piece, will be removed.