My Mac died and I can't access iPhoto on my external hard drive
Background

iPhoto and external storage

It's true that with a normally running operating system on a Mac, an iPhoto library can be stored on an external drive. 

Normal running of iPhoto will involve file system writes to the iPhoto library and elsewhere. 

Time Machine and the file system

When Time Machine backs up to an HFS Plus file system, most of the files that are associated with its backups database are saved in a way that prevents further edition; that prevents deletion – prevention until it becomes necessary for a point in time in the backup to be thinned. 

iPhoto and Time Machine backup databases

Because edition and writes are prevented, iPhoto will be unable to launch and work with any iPhoto library that's within a backup database …



Best use of the backup drive when one of two Macs has failed

Essentially, as soon as possible, you should restore your iPhoto library – and whatever else you will need: 


from the backup drive
to a separate drive


The destination drive can be internal or external. 

Side note: do not aim to restore from the backup drive to a different area of the same drive; you would continue to have a single point of failure, which is horribly risky. 

If the only failure of the failed Mac was its internal disk drive

Consider replacing that failed internal drive. Then use Time Machine to restore data to the replacement drive. 

(What model is the failed Mac, how much memory does it have, and so on? Edit your question to include relevant details – thanks.)

If the internal disk drive of the failed Mac is a good drive, and if that Mac is beyond economical repair

Take the good internal drive. Consider arranging for it to be placed in an enclosure that will be suitable for use with your good Mac laptop. So you'll have an additional external drive for use with whatever computer you want. 

(What model is that good Mac laptop, how much memory does it have, and so on?)

Cost comparison

It's likely that a suitable external drive enclosure will be more economical than a drive. 

Initial cost should be not the only factor in your decision … adding background information (specifications etc.) to your question should help people to make recommendations in answers.