It's very hard to promise that this is quite right, because there are many distinctive features of your preferred style (quite apart from those you draw attention to) which I may not have picked up, and you don't give us sample data to work from. So you will need to check a sample of typical types of literature (e.g., usually, at least a book, an article, a paper in a collection, and in each case with and without DOI and URL information) an may need to refine. Your example also shows some colouring, but it's not clear whether that matters to you.
All that being said, I think the nature style gets you very close, and it's best to start from that and then modify it. It's a style available in TeXlive, so I am assuming it is on Sharelatex, which I don't use.
The modifications are then easy:
To get superscript citations, compressed and sorted (as your example is), simply use \autocite (or \supercite), which is standard \autocite command for the nature style.
To lose the titles of journal articles, pass biblatex the option articletitle=false.
To substitute labels in brackets for the default in nature which is a label with a dot, add \Declarefieldformat{labelnumberwidth}{\mkbibbrackets{#1}} in your preamble.
With those three changes, and using some samples from the default example .bib file, I get the following, which seems pretty close to what you have specified.
EDITED TO ADD (As an object lesson in why these things require precision, having got this I realised a problem: nature uses last-first names, whereas your examples use first-last. To achieve that, we need to make one further modification to the name format, shown in the code below.)
I haven't tested it on Sharelatex can't see any reason why it should't work in any reasonably standard TeX setup.

Full source (amended in light of comment below) for that:
\documentclass{article}
\usepackage[style=nature,articletitle=false]{biblatex}
\addbibresource{biblatex-examples.bib}
\DeclareFieldFormat{labelnumberwidth}{\mkbibbrackets{#1}}
\DeclareNameAlias{default}{given-family}
\begin{document}
A citation\autocite{bertram,springer,cotton}
\printbibliography
\end{document}