The only Chessbase software I'm aware of that might work on the Kindle Fire is the Online Database app. It'll download and play through pgn files from Chessbases's database. There's no support for any sort of ebook right now.

The best free chess app is probably DroidFish. It's based on Stockfish and plays a pretty strong game. I prefer Chess Genius though. It's $4.99 or so.

There's a SCID database viewer available that'll let you read but not modify a SCID games database. I believe someone has taken all the positions from the old Reinfeld tactics books and put them out as a PGN file. I haven't gotten around to finding and trying them out yet.

The Android App for Chess.com will let you play games and use their tactics server (there's a fee for unlimited use, I think). 

There's also TapChess Tactics which costs $1.30 for 100 puzzles, but might be just what you're looking for.

Edit: Both ChessGenius and DroidFish will display pgn files, but have their own quirks. SCID needs files in its format.

Two More: iChess is collection of tactics problems that looks and works great on the Kindle Fire.

Chess Puzzler says it contains 4000 chess problems, not playable, diagrams only. I haven't tried this one yet.