This is a rather hard question, the only script I know of that was deciphered without a bilingue is Linear B, and that was possible only because it turned out that Linear B was used to write Ancient Greek.
Let's get the negatives out first:
I believe this is truly impossible if the language is written in an idiographic script, or if you don't have a lot of written material.
Even then, it will obviously be impossible to discern the language's phonology; at best, if the script is alphabetic, you might be able to find out which letters are consonants and which are vowels/syllabic consonants, if you're lucky and the language has an easy syllable structure.
Also, you very very likely won't find out most of the vocabulary. At best, if you find a word commonly next to a picture of a tree, you could assume the word actually does mean "tree", and try to deduce the meanings of a few other words from that? (Or, as Chadwick did with Linear B, guess that a word is the name of a city) (Or, as Champollion did with Hieroglyphics, guess that a word is the name of a ruler/pharaoh, because it's written in a special way and repeats across documents)
Now, onto the positives though:
What might be your best bet to find out is actually the grammar of the unknown language! (Though still only in tiny bits) If your language is written in an alphabet or abjad, you got the jackpot, because then you can with super high probability (given enough material) identify common affixes in words, and likely identify different sets of words which use different kinds of affixes, which allows you to distinguish word classes. Given that verbs are usually rarer in sentences than nouns (English has a lot of helper verbs, but other languages don't necessarily), and particles are usually rather short, you can even identify word classes.
If you have a language written in a syllabary, this might well be possible as well, but will be harder. (Worse, though also better sometimes, the syllables might stretch across multiple parts of speech; see again the decipherment of Linear B, which relied critically on Alice Kober's insight that exactly this was happening, allowing her to identify syllables with common consonants/vowel contents, but afaik that only enables Chadwick's identification of some words as city names due to repeated consonants/vowels, e.g. ��� as Ko-No-So=Knossos)
Of course, all this is way way way way way WAY more complicated than this, might well not work at all, and is probably impossible if the language is polysynthetic or the script doesn't include spaces or the script is any more complex than just "alphabet" or just "syllabary". But at least, I hope I could give you some scope on the problem.