There are several ways to go about this, but my primary advice would be to concentrate on one thing.
For example, study the instrumentation (how is a specific chord divided between instruments? In what registers are the instruments playing? In which dynamics/articulations/techniques? etc.) - and this doesn't only apply to orchestral works! Even "just" a string quartet can be very enlightening in this aspect.
Or study the form. If it's a sonata form (such as most first movements of classical string quartets), read up on that form, then try finding the specific parts of said form in that piece. That alone can be very challenging in many of Beethoven's later works - so I'd start out with earlier ones.
Or, instead of looking at the broad form, do the contrary and only look at a very small part of the piece, but study it very closely, say, in regards to harmony etc.
There are tons of different approaches, most of which are as valid as the others, if you actually concentrate on them and take them seriously.
Last but not least, it's a great exercise to simply read a score, without actually analysing, and imagining how it might sound. Alternatively: Try playing it on a piano. If it's too hard to play all of it, try reducing it to something you -can- play, such as just the melody and the harmonies below it and a rough idea of the figurations and rhythms. Just the act of reducing a piece to a playable form on the piano can give you huge insights in the things that define a piece. Or, another related alternative: Try conducting the score to an imaginary group of musicians. Look through it carefully, try to see what the critical parts may be and how you would show different aspects of the piece to your performers.
But again: Just be inventive. Just working through score after score with a predetermined method may be useful to some limited degree - but it's often a lot more enlightening to approach it a little bit more "playfully" (which doesn't mean non-seriously...) and challenging yourself to find a new perspective on every new piece you study.
And again: Learn to limit yourself. Don't try to "understand everything" about every piece you're looking at. Set yourself specific, limited goals and go for them.